arX

iv0

912

2137

v2 [

hep-

th]

18 F

eb 2

010

IPhT-t09150

Matrix Models as CFT Genus Expansion

Ivan Kostov1

Institut de Physique Theorique CNRS-URA 2306CEA-Saclay

F-91191 Gif-sur-Yvette France

We show how the formulation of the matrix models as conformalfield theories on a Riemann sur-faces can be used to compute the genus expansion of the observables Here we consider the simplestexample of the hermitian matrix model where the classical solution is described by a hyperellipticRiemann surface To each branch point of the Riemann surfacewe associate an operator which rep-resents a twist field dressed by the modes of the twisted boson The partition function of the matrixmodel is computed as a correlation function of such dressed twist fields The perturbative constructionof the dressing operators yields a set of Feynman rules for the genus expansion which involve ver-tices propagators and tadpoles The vertices are universal the propagators and the tadpoles dependon the the Riemann surface As a demonstration we evaluate the genus-two free energy using theFeynman rules

1Associate member of the Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences 72Tsarigradsko Chaussee 1784 Sofia Bulgaria

1 Introduction

In this paper which is a continuation of [1] we formulate the1N expansion ofU(N) invariantmatrix integrals as the quasiclassical expansion in conformal field theories on Riemann surfaces Ourbasic example will be the hermitian matrix model defined by the partition function

ZN [V ] =

int

dM eminustr V (M)

simint

C

Nprod

i=1

dzi eminusV (zi)

prod

iltj

(zi minus zj)2 (11)

Herez1 zN are the eigenvalues of the matrix variableM and

V (z) = minusinfinsum

n=0

tnzn (12)

is a confining potential For the hermitian matrix integral the contour of integration goes along thereal axis Then the integral is convergent if the potential becomes infinitely large whenz rarr plusmninfin Ifthis is not the case the integral can be made convergent by deforming the contour of integration

The1N expansion of the free energyFN [V ] = lnZN [V ] is of the form

FN [V ] =

infinsum

g=0

N2minus2gF (g)[VN ] + nonperturbative terms (13)

where the termF (g) counts the number of genusg fat Feynman graphs2 The leading genus zeroterm can be obtained by evaluating the saddle point eigenvalue distribution [2] In this approximationthe eigenvalues can be considered as a continuous charged liquid defined by the spectral density

ρ(z) = limNrarrinfin

1

N

Nsum

i=1

δ(z minus zi) (14)

The collective field theory for the spectral density (14) is ill defined at small distances and cannotbe used to compute the higher terms of the quasiclassical expansion This can be done by solving theWard identities for the integral (11) known also as loop equations An improved iterative schemefor calculating the higher genus observables known as method of moments was set up by Amb-jorn Chekhov Kristjansen and Makeenko [3] More recently Eynard [4] formulated the recursionprocedure for solving the loop equations in an elegant graphical scheme

The form of the loop equations suggests that there is a local field theory hidden behind Indeedthe loop equations can be formulated as Virasoro constraints for the bosonic field

φ(z) =radic2

Nsum

i=1

log (z minus zi)minus1radic2V (z) (15)

Therefore the theory in question must be a two-dimensional chiral CFT An important feature of thisCFT is that the bosonic field develops a large classical expectation valueφc(z) The classical current

2Strictly speaking this is true only in the case of one-cut solutions

1

Jc = partzφc satisfies a quadratic equation the classical Virasoro constraint whose solution defines fora general potential a hyperelliptic Riemann surface with branch pointsa1 a2p placed on the realaxis The set of branch cuts along the segments[a1 a2] [a2pminus1 a2p] coincide with the support ofthe classical spectral density Each branch cut is associated with a local minimum of the potential andis characterized by the filling numberNj equal to the number of the eigenvalues trapped around thisminimum

In the chiral CFT the branch points of the Riemann surface canbe thought of as the result ofapplyingtwistoperators to thesl(2) invariant vacuum The twist operators are however not conformalinvariant In particular the translations change the positions of the branch points In [1] we claimedthat in order to satisfy the conformal Ward identity the twist operators should be dressed by the modesof the twisted boson The dressed twist operators were called in [1] star operators because of theanalogy with the star operators introduced by G Moore in [5] We should stress here that the staroperators in [1] and in [5] are different objects

The problem of computing the genus expansion of the of the observables is thus reduced to theperturbative construction of the operators dressing the twist fields Such dressing operator representsa formal series expansion in the modes of the twisted boson The coefficients of the expansion arerelated to the correlation functions in the Kontsevich model In this sense the proposal made in [1]allows to decompose the hermitian matrix model with an arbitrary potential into2p Kontsevich modelscoupled through the modes of a gaussian field on the hyperelliptic surface Similar decompositionformulas for the hermitian matrix model were suggested in [6 7 8]

The aim of this paper is to present the details of the construction of the dressed twist operators andto set up the Feynman diagram technique for calculating the free energy and the expectation values inall orders on1N

The paper is organized as follows In Section2 we remind the Fock space representation of theintegral (11) in terms of a chiral bosonic fieldφ(z) with Liouville-like interaction given in [1] andderive the loop equations from the conformal Ward identityThis Section helps to make the presen-tation self-consistent and can be skipped by the reader who is familiar with the loop equations In3we give the representation of the leading and the subleadingorders of the free energy in terms of thecorrelation function of2p twist operators In Section4 we give the explicit prescription for dressingthe twist operators and set up the Feynman rules for computing all orders of the genus expansion ThisFeynman diagram technique is different than the diagram technique in [4] In Section5 we computeusing the Feynman rules the genus two free energy and compare with the result of [3]

2 The matrix model as a chiral CFT

21 From Coulomb gas to compactified chiral boson

The Coulomb gas integral (11) can be represented as a Fock space expectation value in a theory of aLiouville-like chiral CFT with the wrong sign of the kineticterm Introduce the holomorphic scalarfield φ(z) with mode expansion atz = infin

φ(z) = q + J0 ln z minussum

n 6=0

Jnnzminusn [Jn Jm] = n δn+m0 [J0 q] = 1 (21)

and left and right Fock vacua defined by

〈0|J0 = q|0〉 = 0 〈0|Jminusn = Jn|0〉 = 0 (n ge 1) (22)

2

The field has operator product expansion

φ(z)φ(zprime) sim ln(z minus zprime) (23)

and give a representation of thesu(2) current algebra

J = 1radic2partφ(z) Jplusmn = eplusmn

radic2φ (24)

Then the partition function (11) is given by the scalar product

ZN = 〈N |eJ [V ] eQ+ |0〉 (25)

where

J [V ] = minus∮

infin

dz

2πiV (z)J(z) =

sum

nge0

tn Jn (26)

Q+ =

int infin

minusinfindz J+(z) (27)

and〈N | is the charged left vacuum state

〈N | = 〈0|eradic2Nq (28)

The operatoreQ+ generates screening charges the operatoreJ [V ] produces the measureeminusV (z) foreach charge and the left vacuum〈N | projects onto the sector with exactlyN screening chargesSimilar Fock space representations of the eigenvalue integral have been proposed in [9 10 11]

The currentsJ J+ andJminus are invariant under the discrete translations of of the fieldφ

φ rarr φ+ iπradic2 (29)

Therefore the the fieldφ can be compactified at the self-dual radiusRsd = 1radic2 Furthermore the

transformation

φ rarr minusφ (210)

is an automorphism of the current algebra The geometrical meaning of the symmetry (210) willbecome clear when we consider the quasiclassical limit of the bosonic field

The correlation functions of in the matrix model are obtained through the identification

φ(z) =radic2 tr log (z minusM)minus 1radic

2V (z) (211)

or in terms of thesu(2) currents

J(z) = tr1

z minusMminus 1

2Vprime(z) Jplusmn(z) = e∓V (z)[det(z minusM)]plusmn2 (212)

3

22 Conformal Ward identity

To prove that the theory is conformal invariant we have to demonstrate that the energy-momentumtensor

T (z) = 12 partφ(z)partφ(z) =

sum

n

Lnzminusnminus2 (213)

commutes with the screening operatorQ+ Indeed for anyn ge minus1 we have

[Ln Q+] =

int infin

minusinfindz[Ln J+(z)] =

int infin

minusinfindz

d

dz

(

zn+1J+(z))

(214)

Since our potential diverges at infinity the boundary termsvanish and the result is zero for alln ge minus1As a consequence the expectation value

〈T (z) 〉 def= 〈N |eJ [V ] T (z) eQ+ |0〉 (215)

is regular forz 6= infin This condition can be written as a contour integral which projects to the positivepart of the Laurent expansion ofT (x)

∮

infin

dzprime

2πi

langT (z)minus T (zprime)

z minus zprime

rang

= 0 (216)

The conformal Ward identity (216) is translated into a set of differential Virasoro constraints onthe partition function using the representation of the gaussian field as a differential operator acting onthe partition function

φ(z) rarr φ(z)def=

1radic2

sum

nge0

tnzn +

radic2 ln z

part

partt0+

radic2sum

nge0

zminusn

n

part

parttn (217)

The Virasoro constraints read

Ln middot ZN = 0 (n ge minus1) (218)

where

Lndef=

nsum

k=0

part

parttk

part

parttnminusk+

infinsum

k=0

k tkpart

parttn+k

part

partt0ZN = NZN (219)

3 The quasiclassical limit

31 The classical solution as a hyperelliptic curve

Applied to the genus expansion (13) of the free energy the Virasoro constraints (218) generatean infinite set of equations for the correlation functions ofthe currentJ(z) which can be solvedorder by order in1N The lowest equation is the classical Virasoro condition which determines theexpectation valueJc(z) of the current in the largeN limit In our normalizationJc is of order ofN just as the confining potentialV

4

A AA1 2 3

B1

B2

a a a1 a 3 a4 a5 62

Figure 1 TheA andB canonical cycles for a genus 2 spectral curve(p = 3)

The classical Virasoro constraint states thatTc = J2c is an entire function ofz The most general

solution is

Jc(z) = minusM(z) y(z) y2 =

2pprod

j=1

(z minus aj) (31)

whereM(z) is an entire function ofz Assuming thata1 lt a2 lt lt a2p the meromorphic functionJc(z) is discontinuous along the intervals[a2kminus1 a2k] and its discontinuity is related to the normalizedclassical spectral density by

ρc(x) =Jc(xminus i0) minus Jc(x+ i0)

2πiN (32)

Our aim is to construct a CFT associated with the classical solution Jc It is advantageous to thinkof the gaussian field as defined not on the complex plane cut along the intervals[a2kminus1 a2k] but onthe Riemann surface representing a two-fold branched coverof the complex plane the two sheets ofwhich are sewed along the cuts[a2jminus1 a2j ] In this way we trade the boundary condition along thecuts for the monodromy relations (φ rarr minusφ) when one moves around the branch points

The hyperelliptic Riemann surface is characterized by a setof moduli associated with the canon-ical A andB cycles The cycleAj encircles the cut[a2jminus1 a2j ] and the cycleBj encircles the pointsa2j a2pminus3 passing through thej-th and thep-th cuts (Fig1) so that

Ak Bj = δkj (j = 1 pminus 1) (33)

The classical current is determined completely by the potential V (z) through the asymptotics

Jc(z) = minus12V

prime(z) +Nzminus1 + (34)

and by the chargesN1 + middot middot middot+Np = N associated with theA-cycles∮

Aj

dz

2πiJc(z) = Nj j = 1 p (35)

From (35) it follows that the derivatives

ωj =partJc(z)

partNjdz (36)

5

form a basis of holomorphic differentials of first kind associated with the cyclesAk

1

2πi

∮

Ak

ωj = δkj (37)

The integrals ofωj along theB-cycles give the period matrix of the hyperelliptic curve

τkj =1

2πi

∮

Bk

ωj (38)

In the following we will consider the filling numbersNj as fixed external parameters The cor-responding free energy is that of a metastable state which becomes stable up to exponentially smalleffects in the largeN limit We will understand the1N expansion (13) of the free energy in thissetting Alternatively one can introduce chemical potentials Γj for the chargeNj and evaluate thepartition function for fixedΓj

If one is interested in the quasiclassical evaluation of theoriginal partition function (11) oneshould perform the sum over all possibleNj This problem has been considered and solved in [12]After performing the sum the logarithm of the partition function does not have the1N expansion(13) Since the numbersNj are the discontinuities of the bosonic field along theA-cycles the sumover allNj means that the bosonic field is effectively compactified at the selfdual radius

32 The branch points as primary conformal fields

Our goal is to construct a CFT associated with the classical solution Jc We will think of the hyper-elliptic Riemann surface as the complex plane with conformal operators thetwist operators withdimension 116 associated with the branch points This point of view was first advocated by AlexeiZamolodchikov in [13] Later some of the findings of [13] were obtained independently by Dixonatal [14] and further developed by a series of brilliant papers by V Knizhnik [15 16 17]

Let a be one of the branch points of the classical solutionJc In the vicinity ofa the current hasmode expansion

J(z) = Jc(z) +sum

risinZ+ 12

Jr (z minus a)minusrminus1 (39)

with the following algebra

[Jr Js] =12rδr+s0 (310)

The Ramond vacuum associated with this branch point is defined as the highest weight vector ofthe representation of this algebra The corresponding quantum field is the twist operatorσ(a)

J12σ(a) = J32σ(a) = J52σ(a) = middot middot middot = 0 (311)

The Hilbert space associated withσ(a) generated by multiple action onσ(a) with the negative modesJminus12 Jminus32 of the currentJ

The product of two twist operatorsσ(a1)σ(a2) is a single-valued operator with respect to thecurrentJ(z) This state can be decomposed into charged eigenstates (35) Let us denote a state withgiven chargeN by [σ(a1)σ(a2)]N Our strategy in the following is to simulate the cuts[a2jminus1 a2j ] ofthe Riemann surface by multiplying the right vacuum with theoperators[σ(a2jminus1)σ(a2j)]Nj

and givea Fock-space representation of the partition function of the matrix model similar to (25) In the new

6

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

Jc = partzφc satisfies a quadratic equation the classical Virasoro constraint whose solution defines fora general potential a hyperelliptic Riemann surface with branch pointsa1 a2p placed on the realaxis The set of branch cuts along the segments[a1 a2] [a2pminus1 a2p] coincide with the support ofthe classical spectral density Each branch cut is associated with a local minimum of the potential andis characterized by the filling numberNj equal to the number of the eigenvalues trapped around thisminimum

In the chiral CFT the branch points of the Riemann surface canbe thought of as the result ofapplyingtwistoperators to thesl(2) invariant vacuum The twist operators are however not conformalinvariant In particular the translations change the positions of the branch points In [1] we claimedthat in order to satisfy the conformal Ward identity the twist operators should be dressed by the modesof the twisted boson The dressed twist operators were called in [1] star operators because of theanalogy with the star operators introduced by G Moore in [5] We should stress here that the staroperators in [1] and in [5] are different objects

The problem of computing the genus expansion of the of the observables is thus reduced to theperturbative construction of the operators dressing the twist fields Such dressing operator representsa formal series expansion in the modes of the twisted boson The coefficients of the expansion arerelated to the correlation functions in the Kontsevich model In this sense the proposal made in [1]allows to decompose the hermitian matrix model with an arbitrary potential into2p Kontsevich modelscoupled through the modes of a gaussian field on the hyperelliptic surface Similar decompositionformulas for the hermitian matrix model were suggested in [6 7 8]

The aim of this paper is to present the details of the construction of the dressed twist operators andto set up the Feynman diagram technique for calculating the free energy and the expectation values inall orders on1N

The paper is organized as follows In Section2 we remind the Fock space representation of theintegral (11) in terms of a chiral bosonic fieldφ(z) with Liouville-like interaction given in [1] andderive the loop equations from the conformal Ward identityThis Section helps to make the presen-tation self-consistent and can be skipped by the reader who is familiar with the loop equations In3we give the representation of the leading and the subleadingorders of the free energy in terms of thecorrelation function of2p twist operators In Section4 we give the explicit prescription for dressingthe twist operators and set up the Feynman rules for computing all orders of the genus expansion ThisFeynman diagram technique is different than the diagram technique in [4] In Section5 we computeusing the Feynman rules the genus two free energy and compare with the result of [3]

2 The matrix model as a chiral CFT

21 From Coulomb gas to compactified chiral boson

The Coulomb gas integral (11) can be represented as a Fock space expectation value in a theory of aLiouville-like chiral CFT with the wrong sign of the kineticterm Introduce the holomorphic scalarfield φ(z) with mode expansion atz = infin

φ(z) = q + J0 ln z minussum

n 6=0

Jnnzminusn [Jn Jm] = n δn+m0 [J0 q] = 1 (21)

and left and right Fock vacua defined by

〈0|J0 = q|0〉 = 0 〈0|Jminusn = Jn|0〉 = 0 (n ge 1) (22)

2

The field has operator product expansion

φ(z)φ(zprime) sim ln(z minus zprime) (23)

and give a representation of thesu(2) current algebra

J = 1radic2partφ(z) Jplusmn = eplusmn

radic2φ (24)

Then the partition function (11) is given by the scalar product

ZN = 〈N |eJ [V ] eQ+ |0〉 (25)

where

J [V ] = minus∮

infin

dz

2πiV (z)J(z) =

sum

nge0

tn Jn (26)

Q+ =

int infin

minusinfindz J+(z) (27)

and〈N | is the charged left vacuum state

〈N | = 〈0|eradic2Nq (28)

The operatoreQ+ generates screening charges the operatoreJ [V ] produces the measureeminusV (z) foreach charge and the left vacuum〈N | projects onto the sector with exactlyN screening chargesSimilar Fock space representations of the eigenvalue integral have been proposed in [9 10 11]

The currentsJ J+ andJminus are invariant under the discrete translations of of the fieldφ

φ rarr φ+ iπradic2 (29)

Therefore the the fieldφ can be compactified at the self-dual radiusRsd = 1radic2 Furthermore the

transformation

φ rarr minusφ (210)

is an automorphism of the current algebra The geometrical meaning of the symmetry (210) willbecome clear when we consider the quasiclassical limit of the bosonic field

The correlation functions of in the matrix model are obtained through the identification

φ(z) =radic2 tr log (z minusM)minus 1radic

2V (z) (211)

or in terms of thesu(2) currents

J(z) = tr1

z minusMminus 1

2Vprime(z) Jplusmn(z) = e∓V (z)[det(z minusM)]plusmn2 (212)

3

22 Conformal Ward identity

To prove that the theory is conformal invariant we have to demonstrate that the energy-momentumtensor

T (z) = 12 partφ(z)partφ(z) =

sum

n

Lnzminusnminus2 (213)

commutes with the screening operatorQ+ Indeed for anyn ge minus1 we have

[Ln Q+] =

int infin

minusinfindz[Ln J+(z)] =

int infin

minusinfindz

d

dz

(

zn+1J+(z))

(214)

Since our potential diverges at infinity the boundary termsvanish and the result is zero for alln ge minus1As a consequence the expectation value

〈T (z) 〉 def= 〈N |eJ [V ] T (z) eQ+ |0〉 (215)

is regular forz 6= infin This condition can be written as a contour integral which projects to the positivepart of the Laurent expansion ofT (x)

∮

infin

dzprime

2πi

langT (z)minus T (zprime)

z minus zprime

rang

= 0 (216)

The conformal Ward identity (216) is translated into a set of differential Virasoro constraints onthe partition function using the representation of the gaussian field as a differential operator acting onthe partition function

φ(z) rarr φ(z)def=

1radic2

sum

nge0

tnzn +

radic2 ln z

part

partt0+

radic2sum

nge0

zminusn

n

part

parttn (217)

The Virasoro constraints read

Ln middot ZN = 0 (n ge minus1) (218)

where

Lndef=

nsum

k=0

part

parttk

part

parttnminusk+

infinsum

k=0

k tkpart

parttn+k

part

partt0ZN = NZN (219)

3 The quasiclassical limit

31 The classical solution as a hyperelliptic curve

Applied to the genus expansion (13) of the free energy the Virasoro constraints (218) generatean infinite set of equations for the correlation functions ofthe currentJ(z) which can be solvedorder by order in1N The lowest equation is the classical Virasoro condition which determines theexpectation valueJc(z) of the current in the largeN limit In our normalizationJc is of order ofN just as the confining potentialV

4

A AA1 2 3

B1

B2

a a a1 a 3 a4 a5 62

Figure 1 TheA andB canonical cycles for a genus 2 spectral curve(p = 3)

The classical Virasoro constraint states thatTc = J2c is an entire function ofz The most general

solution is

Jc(z) = minusM(z) y(z) y2 =

2pprod

j=1

(z minus aj) (31)

whereM(z) is an entire function ofz Assuming thata1 lt a2 lt lt a2p the meromorphic functionJc(z) is discontinuous along the intervals[a2kminus1 a2k] and its discontinuity is related to the normalizedclassical spectral density by

ρc(x) =Jc(xminus i0) minus Jc(x+ i0)

2πiN (32)

Our aim is to construct a CFT associated with the classical solution Jc It is advantageous to thinkof the gaussian field as defined not on the complex plane cut along the intervals[a2kminus1 a2k] but onthe Riemann surface representing a two-fold branched coverof the complex plane the two sheets ofwhich are sewed along the cuts[a2jminus1 a2j ] In this way we trade the boundary condition along thecuts for the monodromy relations (φ rarr minusφ) when one moves around the branch points

The hyperelliptic Riemann surface is characterized by a setof moduli associated with the canon-ical A andB cycles The cycleAj encircles the cut[a2jminus1 a2j ] and the cycleBj encircles the pointsa2j a2pminus3 passing through thej-th and thep-th cuts (Fig1) so that

Ak Bj = δkj (j = 1 pminus 1) (33)

The classical current is determined completely by the potential V (z) through the asymptotics

Jc(z) = minus12V

prime(z) +Nzminus1 + (34)

and by the chargesN1 + middot middot middot+Np = N associated with theA-cycles∮

Aj

dz

2πiJc(z) = Nj j = 1 p (35)

From (35) it follows that the derivatives

ωj =partJc(z)

partNjdz (36)

5

form a basis of holomorphic differentials of first kind associated with the cyclesAk

1

2πi

∮

Ak

ωj = δkj (37)

The integrals ofωj along theB-cycles give the period matrix of the hyperelliptic curve

τkj =1

2πi

∮

Bk

ωj (38)

In the following we will consider the filling numbersNj as fixed external parameters The cor-responding free energy is that of a metastable state which becomes stable up to exponentially smalleffects in the largeN limit We will understand the1N expansion (13) of the free energy in thissetting Alternatively one can introduce chemical potentials Γj for the chargeNj and evaluate thepartition function for fixedΓj

If one is interested in the quasiclassical evaluation of theoriginal partition function (11) oneshould perform the sum over all possibleNj This problem has been considered and solved in [12]After performing the sum the logarithm of the partition function does not have the1N expansion(13) Since the numbersNj are the discontinuities of the bosonic field along theA-cycles the sumover allNj means that the bosonic field is effectively compactified at the selfdual radius

32 The branch points as primary conformal fields

Our goal is to construct a CFT associated with the classical solution Jc We will think of the hyper-elliptic Riemann surface as the complex plane with conformal operators thetwist operators withdimension 116 associated with the branch points This point of view was first advocated by AlexeiZamolodchikov in [13] Later some of the findings of [13] were obtained independently by Dixonatal [14] and further developed by a series of brilliant papers by V Knizhnik [15 16 17]

Let a be one of the branch points of the classical solutionJc In the vicinity ofa the current hasmode expansion

J(z) = Jc(z) +sum

risinZ+ 12

Jr (z minus a)minusrminus1 (39)

with the following algebra

[Jr Js] =12rδr+s0 (310)

The Ramond vacuum associated with this branch point is defined as the highest weight vector ofthe representation of this algebra The corresponding quantum field is the twist operatorσ(a)

J12σ(a) = J32σ(a) = J52σ(a) = middot middot middot = 0 (311)

The Hilbert space associated withσ(a) generated by multiple action onσ(a) with the negative modesJminus12 Jminus32 of the currentJ

The product of two twist operatorsσ(a1)σ(a2) is a single-valued operator with respect to thecurrentJ(z) This state can be decomposed into charged eigenstates (35) Let us denote a state withgiven chargeN by [σ(a1)σ(a2)]N Our strategy in the following is to simulate the cuts[a2jminus1 a2j ] ofthe Riemann surface by multiplying the right vacuum with theoperators[σ(a2jminus1)σ(a2j)]Nj

and givea Fock-space representation of the partition function of the matrix model similar to (25) In the new

6

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

The field has operator product expansion

φ(z)φ(zprime) sim ln(z minus zprime) (23)

and give a representation of thesu(2) current algebra

J = 1radic2partφ(z) Jplusmn = eplusmn

radic2φ (24)

Then the partition function (11) is given by the scalar product

ZN = 〈N |eJ [V ] eQ+ |0〉 (25)

where

J [V ] = minus∮

infin

dz

2πiV (z)J(z) =

sum

nge0

tn Jn (26)

Q+ =

int infin

minusinfindz J+(z) (27)

and〈N | is the charged left vacuum state

〈N | = 〈0|eradic2Nq (28)

The operatoreQ+ generates screening charges the operatoreJ [V ] produces the measureeminusV (z) foreach charge and the left vacuum〈N | projects onto the sector with exactlyN screening chargesSimilar Fock space representations of the eigenvalue integral have been proposed in [9 10 11]

The currentsJ J+ andJminus are invariant under the discrete translations of of the fieldφ

φ rarr φ+ iπradic2 (29)

Therefore the the fieldφ can be compactified at the self-dual radiusRsd = 1radic2 Furthermore the

transformation

φ rarr minusφ (210)

is an automorphism of the current algebra The geometrical meaning of the symmetry (210) willbecome clear when we consider the quasiclassical limit of the bosonic field

The correlation functions of in the matrix model are obtained through the identification

φ(z) =radic2 tr log (z minusM)minus 1radic

2V (z) (211)

or in terms of thesu(2) currents

J(z) = tr1

z minusMminus 1

2Vprime(z) Jplusmn(z) = e∓V (z)[det(z minusM)]plusmn2 (212)

3

22 Conformal Ward identity

To prove that the theory is conformal invariant we have to demonstrate that the energy-momentumtensor

T (z) = 12 partφ(z)partφ(z) =

sum

n

Lnzminusnminus2 (213)

commutes with the screening operatorQ+ Indeed for anyn ge minus1 we have

[Ln Q+] =

int infin

minusinfindz[Ln J+(z)] =

int infin

minusinfindz

d

dz

(

zn+1J+(z))

(214)

Since our potential diverges at infinity the boundary termsvanish and the result is zero for alln ge minus1As a consequence the expectation value

〈T (z) 〉 def= 〈N |eJ [V ] T (z) eQ+ |0〉 (215)

is regular forz 6= infin This condition can be written as a contour integral which projects to the positivepart of the Laurent expansion ofT (x)

∮

infin

dzprime

2πi

langT (z)minus T (zprime)

z minus zprime

rang

= 0 (216)

The conformal Ward identity (216) is translated into a set of differential Virasoro constraints onthe partition function using the representation of the gaussian field as a differential operator acting onthe partition function

φ(z) rarr φ(z)def=

1radic2

sum

nge0

tnzn +

radic2 ln z

part

partt0+

radic2sum

nge0

zminusn

n

part

parttn (217)

The Virasoro constraints read

Ln middot ZN = 0 (n ge minus1) (218)

where

Lndef=

nsum

k=0

part

parttk

part

parttnminusk+

infinsum

k=0

k tkpart

parttn+k

part

partt0ZN = NZN (219)

3 The quasiclassical limit

31 The classical solution as a hyperelliptic curve

Applied to the genus expansion (13) of the free energy the Virasoro constraints (218) generatean infinite set of equations for the correlation functions ofthe currentJ(z) which can be solvedorder by order in1N The lowest equation is the classical Virasoro condition which determines theexpectation valueJc(z) of the current in the largeN limit In our normalizationJc is of order ofN just as the confining potentialV

4

A AA1 2 3

B1

B2

a a a1 a 3 a4 a5 62

Figure 1 TheA andB canonical cycles for a genus 2 spectral curve(p = 3)

The classical Virasoro constraint states thatTc = J2c is an entire function ofz The most general

solution is

Jc(z) = minusM(z) y(z) y2 =

2pprod

j=1

(z minus aj) (31)

whereM(z) is an entire function ofz Assuming thata1 lt a2 lt lt a2p the meromorphic functionJc(z) is discontinuous along the intervals[a2kminus1 a2k] and its discontinuity is related to the normalizedclassical spectral density by

ρc(x) =Jc(xminus i0) minus Jc(x+ i0)

2πiN (32)

Our aim is to construct a CFT associated with the classical solution Jc It is advantageous to thinkof the gaussian field as defined not on the complex plane cut along the intervals[a2kminus1 a2k] but onthe Riemann surface representing a two-fold branched coverof the complex plane the two sheets ofwhich are sewed along the cuts[a2jminus1 a2j ] In this way we trade the boundary condition along thecuts for the monodromy relations (φ rarr minusφ) when one moves around the branch points

The hyperelliptic Riemann surface is characterized by a setof moduli associated with the canon-ical A andB cycles The cycleAj encircles the cut[a2jminus1 a2j ] and the cycleBj encircles the pointsa2j a2pminus3 passing through thej-th and thep-th cuts (Fig1) so that

Ak Bj = δkj (j = 1 pminus 1) (33)

The classical current is determined completely by the potential V (z) through the asymptotics

Jc(z) = minus12V

prime(z) +Nzminus1 + (34)

and by the chargesN1 + middot middot middot+Np = N associated with theA-cycles∮

Aj

dz

2πiJc(z) = Nj j = 1 p (35)

From (35) it follows that the derivatives

ωj =partJc(z)

partNjdz (36)

5

form a basis of holomorphic differentials of first kind associated with the cyclesAk

1

2πi

∮

Ak

ωj = δkj (37)

The integrals ofωj along theB-cycles give the period matrix of the hyperelliptic curve

τkj =1

2πi

∮

Bk

ωj (38)

In the following we will consider the filling numbersNj as fixed external parameters The cor-responding free energy is that of a metastable state which becomes stable up to exponentially smalleffects in the largeN limit We will understand the1N expansion (13) of the free energy in thissetting Alternatively one can introduce chemical potentials Γj for the chargeNj and evaluate thepartition function for fixedΓj

If one is interested in the quasiclassical evaluation of theoriginal partition function (11) oneshould perform the sum over all possibleNj This problem has been considered and solved in [12]After performing the sum the logarithm of the partition function does not have the1N expansion(13) Since the numbersNj are the discontinuities of the bosonic field along theA-cycles the sumover allNj means that the bosonic field is effectively compactified at the selfdual radius

32 The branch points as primary conformal fields

Our goal is to construct a CFT associated with the classical solution Jc We will think of the hyper-elliptic Riemann surface as the complex plane with conformal operators thetwist operators withdimension 116 associated with the branch points This point of view was first advocated by AlexeiZamolodchikov in [13] Later some of the findings of [13] were obtained independently by Dixonatal [14] and further developed by a series of brilliant papers by V Knizhnik [15 16 17]

Let a be one of the branch points of the classical solutionJc In the vicinity ofa the current hasmode expansion

J(z) = Jc(z) +sum

risinZ+ 12

Jr (z minus a)minusrminus1 (39)

with the following algebra

[Jr Js] =12rδr+s0 (310)

The Ramond vacuum associated with this branch point is defined as the highest weight vector ofthe representation of this algebra The corresponding quantum field is the twist operatorσ(a)

J12σ(a) = J32σ(a) = J52σ(a) = middot middot middot = 0 (311)

The Hilbert space associated withσ(a) generated by multiple action onσ(a) with the negative modesJminus12 Jminus32 of the currentJ

The product of two twist operatorsσ(a1)σ(a2) is a single-valued operator with respect to thecurrentJ(z) This state can be decomposed into charged eigenstates (35) Let us denote a state withgiven chargeN by [σ(a1)σ(a2)]N Our strategy in the following is to simulate the cuts[a2jminus1 a2j ] ofthe Riemann surface by multiplying the right vacuum with theoperators[σ(a2jminus1)σ(a2j)]Nj

and givea Fock-space representation of the partition function of the matrix model similar to (25) In the new

6

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

22 Conformal Ward identity

To prove that the theory is conformal invariant we have to demonstrate that the energy-momentumtensor

T (z) = 12 partφ(z)partφ(z) =

sum

n

Lnzminusnminus2 (213)

commutes with the screening operatorQ+ Indeed for anyn ge minus1 we have

[Ln Q+] =

int infin

minusinfindz[Ln J+(z)] =

int infin

minusinfindz

d

dz

(

zn+1J+(z))

(214)

Since our potential diverges at infinity the boundary termsvanish and the result is zero for alln ge minus1As a consequence the expectation value

〈T (z) 〉 def= 〈N |eJ [V ] T (z) eQ+ |0〉 (215)

is regular forz 6= infin This condition can be written as a contour integral which projects to the positivepart of the Laurent expansion ofT (x)

∮

infin

dzprime

2πi

langT (z)minus T (zprime)

z minus zprime

rang

= 0 (216)

The conformal Ward identity (216) is translated into a set of differential Virasoro constraints onthe partition function using the representation of the gaussian field as a differential operator acting onthe partition function

φ(z) rarr φ(z)def=

1radic2

sum

nge0

tnzn +

radic2 ln z

part

partt0+

radic2sum

nge0

zminusn

n

part

parttn (217)

The Virasoro constraints read

Ln middot ZN = 0 (n ge minus1) (218)

where

Lndef=

nsum

k=0

part

parttk

part

parttnminusk+

infinsum

k=0

k tkpart

parttn+k

part

partt0ZN = NZN (219)

3 The quasiclassical limit

31 The classical solution as a hyperelliptic curve

Applied to the genus expansion (13) of the free energy the Virasoro constraints (218) generatean infinite set of equations for the correlation functions ofthe currentJ(z) which can be solvedorder by order in1N The lowest equation is the classical Virasoro condition which determines theexpectation valueJc(z) of the current in the largeN limit In our normalizationJc is of order ofN just as the confining potentialV

4

A AA1 2 3

B1

B2

a a a1 a 3 a4 a5 62

Figure 1 TheA andB canonical cycles for a genus 2 spectral curve(p = 3)

The classical Virasoro constraint states thatTc = J2c is an entire function ofz The most general

solution is

Jc(z) = minusM(z) y(z) y2 =

2pprod

j=1

(z minus aj) (31)

whereM(z) is an entire function ofz Assuming thata1 lt a2 lt lt a2p the meromorphic functionJc(z) is discontinuous along the intervals[a2kminus1 a2k] and its discontinuity is related to the normalizedclassical spectral density by

ρc(x) =Jc(xminus i0) minus Jc(x+ i0)

2πiN (32)

Our aim is to construct a CFT associated with the classical solution Jc It is advantageous to thinkof the gaussian field as defined not on the complex plane cut along the intervals[a2kminus1 a2k] but onthe Riemann surface representing a two-fold branched coverof the complex plane the two sheets ofwhich are sewed along the cuts[a2jminus1 a2j ] In this way we trade the boundary condition along thecuts for the monodromy relations (φ rarr minusφ) when one moves around the branch points

The hyperelliptic Riemann surface is characterized by a setof moduli associated with the canon-ical A andB cycles The cycleAj encircles the cut[a2jminus1 a2j ] and the cycleBj encircles the pointsa2j a2pminus3 passing through thej-th and thep-th cuts (Fig1) so that

Ak Bj = δkj (j = 1 pminus 1) (33)

The classical current is determined completely by the potential V (z) through the asymptotics

Jc(z) = minus12V

prime(z) +Nzminus1 + (34)

and by the chargesN1 + middot middot middot+Np = N associated with theA-cycles∮

Aj

dz

2πiJc(z) = Nj j = 1 p (35)

From (35) it follows that the derivatives

ωj =partJc(z)

partNjdz (36)

5

form a basis of holomorphic differentials of first kind associated with the cyclesAk

1

2πi

∮

Ak

ωj = δkj (37)

The integrals ofωj along theB-cycles give the period matrix of the hyperelliptic curve

τkj =1

2πi

∮

Bk

ωj (38)

In the following we will consider the filling numbersNj as fixed external parameters The cor-responding free energy is that of a metastable state which becomes stable up to exponentially smalleffects in the largeN limit We will understand the1N expansion (13) of the free energy in thissetting Alternatively one can introduce chemical potentials Γj for the chargeNj and evaluate thepartition function for fixedΓj

If one is interested in the quasiclassical evaluation of theoriginal partition function (11) oneshould perform the sum over all possibleNj This problem has been considered and solved in [12]After performing the sum the logarithm of the partition function does not have the1N expansion(13) Since the numbersNj are the discontinuities of the bosonic field along theA-cycles the sumover allNj means that the bosonic field is effectively compactified at the selfdual radius

32 The branch points as primary conformal fields

Our goal is to construct a CFT associated with the classical solution Jc We will think of the hyper-elliptic Riemann surface as the complex plane with conformal operators thetwist operators withdimension 116 associated with the branch points This point of view was first advocated by AlexeiZamolodchikov in [13] Later some of the findings of [13] were obtained independently by Dixonatal [14] and further developed by a series of brilliant papers by V Knizhnik [15 16 17]

Let a be one of the branch points of the classical solutionJc In the vicinity ofa the current hasmode expansion

J(z) = Jc(z) +sum

risinZ+ 12

Jr (z minus a)minusrminus1 (39)

with the following algebra

[Jr Js] =12rδr+s0 (310)

The Ramond vacuum associated with this branch point is defined as the highest weight vector ofthe representation of this algebra The corresponding quantum field is the twist operatorσ(a)

J12σ(a) = J32σ(a) = J52σ(a) = middot middot middot = 0 (311)

The Hilbert space associated withσ(a) generated by multiple action onσ(a) with the negative modesJminus12 Jminus32 of the currentJ

The product of two twist operatorsσ(a1)σ(a2) is a single-valued operator with respect to thecurrentJ(z) This state can be decomposed into charged eigenstates (35) Let us denote a state withgiven chargeN by [σ(a1)σ(a2)]N Our strategy in the following is to simulate the cuts[a2jminus1 a2j ] ofthe Riemann surface by multiplying the right vacuum with theoperators[σ(a2jminus1)σ(a2j)]Nj

and givea Fock-space representation of the partition function of the matrix model similar to (25) In the new

6

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

A AA1 2 3

B1

B2

a a a1 a 3 a4 a5 62

Figure 1 TheA andB canonical cycles for a genus 2 spectral curve(p = 3)

The classical Virasoro constraint states thatTc = J2c is an entire function ofz The most general

solution is

Jc(z) = minusM(z) y(z) y2 =

2pprod

j=1

(z minus aj) (31)

whereM(z) is an entire function ofz Assuming thata1 lt a2 lt lt a2p the meromorphic functionJc(z) is discontinuous along the intervals[a2kminus1 a2k] and its discontinuity is related to the normalizedclassical spectral density by

ρc(x) =Jc(xminus i0) minus Jc(x+ i0)

2πiN (32)

Our aim is to construct a CFT associated with the classical solution Jc It is advantageous to thinkof the gaussian field as defined not on the complex plane cut along the intervals[a2kminus1 a2k] but onthe Riemann surface representing a two-fold branched coverof the complex plane the two sheets ofwhich are sewed along the cuts[a2jminus1 a2j ] In this way we trade the boundary condition along thecuts for the monodromy relations (φ rarr minusφ) when one moves around the branch points

The hyperelliptic Riemann surface is characterized by a setof moduli associated with the canon-ical A andB cycles The cycleAj encircles the cut[a2jminus1 a2j ] and the cycleBj encircles the pointsa2j a2pminus3 passing through thej-th and thep-th cuts (Fig1) so that

Ak Bj = δkj (j = 1 pminus 1) (33)

The classical current is determined completely by the potential V (z) through the asymptotics

Jc(z) = minus12V

prime(z) +Nzminus1 + (34)

and by the chargesN1 + middot middot middot+Np = N associated with theA-cycles∮

Aj

dz

2πiJc(z) = Nj j = 1 p (35)

From (35) it follows that the derivatives

ωj =partJc(z)

partNjdz (36)

5

form a basis of holomorphic differentials of first kind associated with the cyclesAk

1

2πi

∮

Ak

ωj = δkj (37)

The integrals ofωj along theB-cycles give the period matrix of the hyperelliptic curve

τkj =1

2πi

∮

Bk

ωj (38)

In the following we will consider the filling numbersNj as fixed external parameters The cor-responding free energy is that of a metastable state which becomes stable up to exponentially smalleffects in the largeN limit We will understand the1N expansion (13) of the free energy in thissetting Alternatively one can introduce chemical potentials Γj for the chargeNj and evaluate thepartition function for fixedΓj

If one is interested in the quasiclassical evaluation of theoriginal partition function (11) oneshould perform the sum over all possibleNj This problem has been considered and solved in [12]After performing the sum the logarithm of the partition function does not have the1N expansion(13) Since the numbersNj are the discontinuities of the bosonic field along theA-cycles the sumover allNj means that the bosonic field is effectively compactified at the selfdual radius

32 The branch points as primary conformal fields

Our goal is to construct a CFT associated with the classical solution Jc We will think of the hyper-elliptic Riemann surface as the complex plane with conformal operators thetwist operators withdimension 116 associated with the branch points This point of view was first advocated by AlexeiZamolodchikov in [13] Later some of the findings of [13] were obtained independently by Dixonatal [14] and further developed by a series of brilliant papers by V Knizhnik [15 16 17]

Let a be one of the branch points of the classical solutionJc In the vicinity ofa the current hasmode expansion

J(z) = Jc(z) +sum

risinZ+ 12

Jr (z minus a)minusrminus1 (39)

with the following algebra

[Jr Js] =12rδr+s0 (310)

The Ramond vacuum associated with this branch point is defined as the highest weight vector ofthe representation of this algebra The corresponding quantum field is the twist operatorσ(a)

J12σ(a) = J32σ(a) = J52σ(a) = middot middot middot = 0 (311)

The Hilbert space associated withσ(a) generated by multiple action onσ(a) with the negative modesJminus12 Jminus32 of the currentJ

The product of two twist operatorsσ(a1)σ(a2) is a single-valued operator with respect to thecurrentJ(z) This state can be decomposed into charged eigenstates (35) Let us denote a state withgiven chargeN by [σ(a1)σ(a2)]N Our strategy in the following is to simulate the cuts[a2jminus1 a2j ] ofthe Riemann surface by multiplying the right vacuum with theoperators[σ(a2jminus1)σ(a2j)]Nj

and givea Fock-space representation of the partition function of the matrix model similar to (25) In the new

6

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

form a basis of holomorphic differentials of first kind associated with the cyclesAk

1

2πi

∮

Ak

ωj = δkj (37)

The integrals ofωj along theB-cycles give the period matrix of the hyperelliptic curve

τkj =1

2πi

∮

Bk

ωj (38)

In the following we will consider the filling numbersNj as fixed external parameters The cor-responding free energy is that of a metastable state which becomes stable up to exponentially smalleffects in the largeN limit We will understand the1N expansion (13) of the free energy in thissetting Alternatively one can introduce chemical potentials Γj for the chargeNj and evaluate thepartition function for fixedΓj

If one is interested in the quasiclassical evaluation of theoriginal partition function (11) oneshould perform the sum over all possibleNj This problem has been considered and solved in [12]After performing the sum the logarithm of the partition function does not have the1N expansion(13) Since the numbersNj are the discontinuities of the bosonic field along theA-cycles the sumover allNj means that the bosonic field is effectively compactified at the selfdual radius

32 The branch points as primary conformal fields

Our goal is to construct a CFT associated with the classical solution Jc We will think of the hyper-elliptic Riemann surface as the complex plane with conformal operators thetwist operators withdimension 116 associated with the branch points This point of view was first advocated by AlexeiZamolodchikov in [13] Later some of the findings of [13] were obtained independently by Dixonatal [14] and further developed by a series of brilliant papers by V Knizhnik [15 16 17]

Let a be one of the branch points of the classical solutionJc In the vicinity ofa the current hasmode expansion

J(z) = Jc(z) +sum

risinZ+ 12

Jr (z minus a)minusrminus1 (39)

with the following algebra

[Jr Js] =12rδr+s0 (310)

The Ramond vacuum associated with this branch point is defined as the highest weight vector ofthe representation of this algebra The corresponding quantum field is the twist operatorσ(a)

J12σ(a) = J32σ(a) = J52σ(a) = middot middot middot = 0 (311)

The Hilbert space associated withσ(a) generated by multiple action onσ(a) with the negative modesJminus12 Jminus32 of the currentJ

The product of two twist operatorsσ(a1)σ(a2) is a single-valued operator with respect to thecurrentJ(z) This state can be decomposed into charged eigenstates (35) Let us denote a state withgiven chargeN by [σ(a1)σ(a2)]N Our strategy in the following is to simulate the cuts[a2jminus1 a2j ] ofthe Riemann surface by multiplying the right vacuum with theoperators[σ(a2jminus1)σ(a2j)]Nj

and givea Fock-space representation of the partition function of the matrix model similar to (25) In the new

6

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

representation the operatoreQ+ creating theN charges is replaced by a product ofp pairs of twistoperators

The correlation function ofp pairs of twist operators was calculated by Al Zamolodchikov [13]in the case when the current has no expectation value and the total charge is zero

lang

0∣

∣

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

= Ztwist(a1 a2p) eiπ

sump

jk=1 τjkNjNk (312)

The meromorphic functionZtwist(a1 a2p) is given by

Ztwist =

2pprod

jltk

(aj minus ak)minus18 [detK]minus12 (313)

where the matrixK is defined by

Kij =

int

Ai

zjminus1dz

y(z)(i j = 1 pminus 1) (314)

It is straightforward to generalize this formula to the caseof non-zero total charge and non-vanishingexpectation value of the current We should simply insert the operatorexp J [V ] as in (25) This givesa representation of the partition function in the quasiclassical limit as the scalar product

ZquasiclN =

lang

N∣

∣ eJ [V ]pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(315)

The normalized expectation value of the current evaluated with respect to (315) obviously coin-cides with the classical solutionJc

〈 J(z) 〉quasicldef= 1radic

2partφ(z) middot logZquasicl

N = Jc(z) (316)

Therefore the rhs of (315) satisfies the classical Virasoro constraints and reproduces correctly theleading order of the free energy The quantum Virasoro constraints are however not respected Thetwist operatorσ(a) depends on the position of the branch pointa and does not satisfy the lowestVirasoro conditionLminus1 It is also not invariant with respect to dilatations generated byL0

4 The genus expansion

41 Dressed twist operators

We would like to modify (315) so that it reproduces all orders of the1N expansion of the free energyWe have to look for operators which create conformally invariant states near the branch points Suchoperators can be constructed from the modes of the twisted bosonic field near the branch point byrequiring that the singular terms with their OPE with the energy-momentum tensor vanish

In [1] the author proposed3 that twist operatorσ(a) can be made conformal invariant by mul-tiplying with an appropriate dressing operatorew(a) which is made from the modes of the twistedfield

σ(a) rarr S(a) = ew(a)σ(a) (41)

3This proposal was reproduced in more details in sect 43 of [18] following an unpublished extended version of [1]

7

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

Assuming that the dressed twist operator is a well defined operator in the Hilbert space associatedwith σ(a) the dressing exponentw(a) can be expanded as a formal series in the creation operatorsJminusr (r gt 0) defined by the expansion (39) near the pointa

w(a) =sum

nge0

1

n

sum

r1rn

wr1rn(a)Jminusr1 Jminusrn (42)

The coefficients of the expansion depend on the classical current Jc and are determined by the re-quirement of conformal invariance

Ln S(a) = 0 (n ge minus1) (43)

where the Virasoro generatorsLn are defined by the expansion at the pointz = a of the energy-momentum tensor

T (z) = limzprimerarr0

[

J(z)J(zprime)minus 1

2(z minus zprime)2

]

=sum

n

Ln(a) (z minus a)minusnminus2 (44)

One finds for the Virasoro operators withn ge minus1

Ln =sum

r+s=n

(Jr + Jcr ) (Js + Jc

s ) +1

16δn0 (45)

whereJcr are the modes in the expansion of the classical current near the pointz = a

Jc(z) =sum

rge32

Jcminusr (z minus a)rminus1 (46)

By conventionJcminusr = 0 if r le 12

Once we found conformal invariant operators that create thebranch points it is clear how to repair(315) so that it holds for all orders in1N2 It is still possible to assign to the pair of dressed twistoperators associated with the endpoints of the cut[a2jminus1 a2j ] a definite chargeNj We denote thisstate by[S(a2jminus1)S(a2j)]Nj

Our claim is that the partition function of the matrix model is equal up to non-perturbative terms

to the expectation value

ZN =lang

N∣

∣ eJ [V ]pprod

j=1

[S(a2jminus1)S(a2j)]Nj

∣

∣0rang

(47)

whereJ(z) is the current of theZ2-twisted gaussian field defined on the Riemann surface of theclassical solution Indeed this expression satisfies the conformal Ward identity in all orders of1Nsince by construction the energy-momentum tensor (44) commutes with the dressed twist operatorsFurthermore the leading order expectation value of the current Jc(z) satisfies the asymptotics atinfinity (34) and the conditions (35)

Remark The Fock space realization (47) of the partition function resembles the QFT represen-tation of theτ -function for isomonodromic deformations obtained by T Miwa [19] and revisited byG Moore [5] In our case there is an irregular singularity at infinity and 2p regular singularities at thebranch points For matrix ensembles with hard edges (infinite wall potential) it was convincingly ar-gued in [20] that the Fock space representation (25) can be transformed into to a correlation function

8

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

of Moorersquos star operators However even in this simplest case the perturbative or1N expansionof the star operators is ill defined The ambiguity gets even worse for smooth potentials The staroperator is defined in [5] by the exponential of a contour integral starting at the branch point but inthe case of a smooth potential the position of the branch point should be adjusted at each order in1N The dressed twist operators (41) generate the complete perturbative expansion while the staroperators of [5] capture the leading non-perturbative behavior

42 Fock space representation of the partition function

The1N expansion can be obtained by considering the dressing operators as a perturbation and ex-

pand them in the negative modes of the current Let us denote by J[aj ]r the modes of the currentJ

associated with the expansion around the branch pointaj Then the total dressing operator which wedenote byΩ is given by the formal series

Ω =prod

j=12p

ew(aj ) w(aj) =sum

nge0

1

n

sum

r1rn

w[aj ]r1rn J

[aj ]minusr1 J

[aj ]minusrn (48)

The partition functionZN is equal up to non-perturbative corrections to the normalized expectationvalue of the dressing operator with respect to the left and right states

lang

left∣

∣

def=

lang

N∣

∣ eJ [V ]∣

∣rightrang def=

pprod

j=1

[σ(a2jminus1)σ(a2j)]Nj

∣

∣0rang

(49)

In order to perform the expansion we should evaluate the expectation value of any product of

negative modesJ[aj ]r SinceJ(z) is the current of a gaussian field it is sufficient to calculate the

expectation value of a pair of such modes

G[aiaj ]rs = 〈J [ai]

minusr J[aj ]minuss 〉 def

=

lang

left∣

∣J[ai]minusr J

[aj ]minuss

∣

∣rightrang

lang

left∣

∣rightrang (410)

The matrixG(aiaj)rrprime can be computed knowing the two-point function〈J(z)J(zprime) 〉 which is the

unique function defined globally on the Riemann surface and having a double pole atz = zprime withresidue12 From the definition (410) and the mode expansion (39) it follows that

G[aiaj ]rrprime =

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)r(zprime minus aj)r

prime (411)

Once we know the matrixG(aiaj)rs and the coefficientsw

[aj ]r1rn we can compute the1N expansion

to any order just by expanding the dressing operators and performing Wick contractionsThis prescription can be packed in a concise formula in the following way Introduce together

with the right Fock vacuum associated with the2p twist operators a left twisted Fock vacuum whichannihilates the negative modes of the current The two vacuum states which we denote by〈0tw| and|0tw〉 are defined by

〈0tw|J [aj ]minusr = 0 J

[aj ]r |0tw〉 = 0 (r ge 12 j = 1 2p) (412)

Then the state∣

∣rightrang

can be identified with|0tw〉 and the statelang

left∣

∣ is obtained from〈0tw| by actingwith the gaussian operator associated with the matrix (410) As a result we obtain the following Fockspace representation of the expectation value (47)

ZN =lang

left∣

∣Ω∣

∣rightrang

=lang

left∣

∣rightrang

〈0tw| e2JGJ Ω |0tw〉 (413)

9

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

whereΩ is defined by (48) and

JGJdef=

2psum

ij=1

sum

rsge12

1

r sG

[aiaj ]rs J [ai]

r J[aj ]s (414)

43 The dressing operator and the Kontsevich integral

In order to make use of this formula we need the explicit expressions for the coefficients of the series(42) which can be obtained by demanding that the dressing operator Ω = ew solve the Virasoroconstraints generated by the operators (45)

A detailed analysis of the solution of these Virasoro constraints can be found in [21] The solutionis given by the Kontsevich matrix integral [22] also known as matrix Airy function To make theconnection with the notations of [21] we represent the modesJr as4

Jminusnminus12 = minus12tn Jn+12 = minus(n+ 1

2) partn (n ge 0) (415)

where we denotedpartn equiv partparttn We also relabel the modes of the classical current as5

Jcminusnminus 1

2rarr minus1

2micron (n ge 0) (416)

Then the dressing operator (42) is represented by a functionΩ(t0 t1 t2 ) which satisfies

LnΩ = 0 (n ge minus1) (417)

with

Ln =sum

kminusm=n

(k + 12)(tm + microm)partk +

sum

k+m=nminus1

(k + 12)(m+ 1

2 )partkpartm (418)

+1

4t20 δn+10 +

1

16δn0 (n ge minus1) (419)

The solution depends on the momentsmicron of the classical current It is sufficient to have thesolutionΩ0 for the simplest nontrivial classical backgroundmicron = micro1 δn1 Then the general solutionis obtained simply by a shifttn rarr tn minus micron n ge 2

Ω = exp(

minussum

nge2

micronpartn

)

Ω0 (420)

The functionΩ0 is given by a formal expansion int0 t1 and1micro1 sim 1N

Ω0(tn) = micro1minus124 exp

sum

gge0

sum

nge0

sum

k1knge0

micro12minus2gminusn w

(g)k1kn

tk1 tknn

(421)

The coefficientsw(g)k1kn

are the genusg correlation functions in the Kontsevich model They areproportional to the intersection numbers in the moduli space of Riemann surfaces of genusg with npunctures

w(g)k1kn

= (minus1)nnprod

j=1

(2kj minus 1) 〈τk1 τkn〉g (422)

4The timestn here are shifted with respect to the times in [21] as(tn)here= [(2nminus 1)]minus1(tn minus δn1)there5The momentsIn of [21] are related tomicron by I1 = 1minus micro1 andIn = minus(2nminus 1)micron for n ge 2

10

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

The intersection numbers〈τk1 τkn〉g are positive rationals They are nonzero only if the indicesk1 kn obey the selection rule

nsum

j=1

kj = 3(g minus 1) + n (423)

The genus zero intersection numbers are the multinomial coefficients

〈τm1 τmn〉0 =(m1 + +mn)

m1mn m1 + +mn = nminus 3 (424)

The first several intersection numbers of genusg = 1 2 are [21]

〈τn1 〉1 =(nminus 1)

24 〈τn0 τn+1〉1 =

1

24 〈τ0τ1τ2〉1 =

1

12

〈τ32 〉2 =7

240 〈τ2τ3〉2 =

29

5760 〈τ4〉2 =

1

1152 (425)

An efficient procedure for evaluating〈τk1 τkn〉g was proposed in [23]

44 Feynman rules

Now we have all the ingredients needed to construct the1N expansion of the free energy associatedwith the classical solution withp cuts Our starting point is the Fock space representation (413) ofthe all genus partition function The expression for the partition function depends on the moments ofthe classical current at the branch points

micro[aj ]n = minus2

∮

aj

dz

2πi

Jc(z)

(z minus aj)n+12(n ge 1) (426)

and the matrixg[aiaj ]mn associated with the two-point function of the current on theRiemann surface

and defined by

G[aiaj ]mn = 4

int

dz

2πi

int

dzprime

2πi

〈J(z)J(zprime)〉(z minus ai)m+12(zprime minus aj)n+12

(427)

With each branch point we associate a set of coordinatest[aj ]n and use the representation (415)

of the modes of the bosonic current in terms of the Heisenbergalgebra generated byt[aj ]n andpart

[aj ]n

Together with the explicit solution (421) for the dressing operators associated with the branch pointsthis leads to the following expression for the genus expansion of the free energy

eN2F(0)+F(1)

= eiπsump

jk=1 τjkNjNk+sum

n tnJn

2pprod

j=1

(

micro[aj ]1

)minus124Ztwist(a1 a2p) (428)

esum

gge2 N2minus2gF(g)

= exp

12

2psum

ij=1

sum

mnge0

G[aiaj ]mn part[ai]

m part[aj ]n

exp

2psum

j=1

sum

nge0

micro[aj ]n part

[aj ]n

times exp

sum

gge0

sum

nge0

sum

k1knge0

(

micro[aj ]1

)2minus2gminusnw

(g)k1kn

t[aj ]k1

t[aj ]kn

n

t(middot)middot =0

(429)

11

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

ltW(z)gt ltW(z)W(zrsquo)gt

i

i jG nm

a ai j[ ] ai[ ]

ai

aia ajnm

n

mm

m

z

G z( )

m

m

[ ]ia

z zzrsquo

mm

m

ai

ai

nmG[ ]a a

2 n

micro

handlesg

12

n

w(g)m m m1

Figure 2 The Feynman rules for genus expansion of the hermitian matrix model The verticesW (g)m1mn

are universal the propagatorsG[aiaj ]mn depend only on the moduli of the spectral curve and the tadpolesmicro[aj ]

m

are determined by the expansion of the classical solution atthe branch points The vertices are connected toeach other by propagators and the tadpoles are connected directly to the vertices The Feynman graphs for thecorrelation functions of the resolvent contain three more elements the disk the cylinder and the external legsdepicted in the second line

Using this formula one can evaluate the genusg free energy as a sum of a finite number of Feyn-

man graphs with verticesw(g)k1kn

given by (422) tadpolesmicro[aj ]n and propagatorsG

[aiaj ]mn Note that

while the propagators and the tadpoles depend on the classical solution the vertices are universalFrom (425) we find the first several vertices (422)

w(0)000 = minus1 w

(1)1 = minus 1

24 w

(1)11 =

1

24 w

(1)02 = minus1

8 w

(1)012 = minus1

4 w

(1)003 = minus5

8

w(1)0022 =

3

2w

(2)222 = minus63

80 w

(2)23 =

29

128 w

(2)4 = minus 35

384 w

(2)00002 = minus3 (430)

The correlators of the resolvent of the random matrix are obtained from the correlations of thecurrentJ by the identification (212) To evaluate the correlation functions of the current we use itsrepresentation as differential operator

J(z) = Jc(z) +

2psum

j=1

sum

nge0

G[aj ]n (z) part

[aj ]n (431)

We therefore extend the set of Feynman rules by adding the external lines which represent the func-tions

G[aj ]n (z) = 2

int

dzprime

2πi

〈J(z)J(zprime)〉(zprime minus aj)n+12

(432)

The Feynman rues for evaluating the1N expansion are given in Fig2 The genusg contributionfor any observable is equal to the sum of all genusg Feynman diagrams The genus of a Feynmandiagram is equal to2 minus 2h minus n whereh is the number of handles including the handles made ofpropagators andn is the number of the external lines

12

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

5 Example the single cut solution

In this section we consider in details the case of one cut(p = 1) where the Riemann surface of theclassical solution is a sphere The explicit expression forthe classical current is

Jc(z) =12

∮

A1

dz

2πi

y(z)

y(zprime)

V prime(zprime)minus V prime(z)

z minus zprime y(z) =

radic

(z minus a1)(z minus a2) (51)

Expanding atz = infin and using the asymptoticsJc(z) sim minus12V

prime(z)+Nz+ one finds the conditions

∮

A1

dz

2πi

zkV prime(z)

y(z)= minus2Nδk1 (k = 0 1) (52)

which determine the positions of the two branch pointsThe two-point function of theZ2-twisted currentJ(z) on the Riemann surface of the classical

solution is equal to the 4-point function of two currents andtwo twist operators

〈J(z)J(zprime)〉 equiv 〈0|J(z)J(zprime)σ(a)σ(b)|0〉 =

radic

(zminusa1)(zprimeminusa2)(zminusa2)(zprimeminusa1)

+radic

(zprimeminusa1)(zminusa2)(zprimeminusa2)(zminusa1)

4(z minus zprime)2 (53)

The coefficientsG[aiak]km are obtained by expanding〈J(z)J(zprime)〉 near the pointsa1 anda2 Assuming

thata2 gt a1 we write

G[aiaj ]km = 4

∮

ai

dz

2πi

∮

aj

dzj2πi

(z minus ai)minuskminus 1

2 (zprime minus aj)minusmminus 1

2 〈J(z)J(zprime)〉 (54)

We find thatG[a1a1]km are of the form

G[aiak]km =

1

dk+m+1g[aiak ]km d = |a1 minus a2| (55)

whereg[aiaj ]km are rational numbers with the symmetryg[a1a1]km = g

[a2a2]km andg[a1a2]km = g

[a2a1]km The

first several coefficientsg[aiaj ]km are

g(a1 a1)km kmge0

= g(a2 a2)km kmge0

=

minus12 38 minus516 38 minus38 45128

minus516 45128 minus45128

g(a1 a2)km kmge0

= g(a2 a1)km kmge0

=

minus1 32 minus158 32 minus214 16516

minus158 16516 minus174564

(56)

As an illustration of the Feynman diagram technique we will evaluate the free energy up to genustwo We denote the two branch points byaprime = a1 a = a2 and the moments of the classical solution

associated with them bymicron = micro(a)n microprime

n = micro(aprime)n

The genus-one term is

F (1) = minus 1

24lnmicro1 minus

1

24lnmicroprime

1 minus1

8ln d (d = aminus aprime) (57)

13

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

minus12

1

11

1

0

1

0

0

1

minus214

32

0

minus1

minus38

38

00

4

000

22

2

minus1

2200

2

3

29128

2

0

18

1

1

20000

minus3

012

minus14

003

1

minus124

minus35384 minus5832

minus6380

124

Figure 3The verticesw(g)m1mn and the propagatorsg[aa

prime]mn andg[aa]mn = g

[aprimeaprime]mn contributing to the genus two

free energy in the case of a single cut

0

222

2

34

0

00

1

1 0

2

1

1

00

00

00

00

0

00

0

0

1

00

0 0

2 2 2 00

00

2

0 1

110

00 1

00

01

00 0

00

0 0

0 0

00

1

0

2

01

2

2

0000

2

000

00

3

00

2

2

00

2

2

1

minus164 minus132 minus1128

minus7768 minus164 minus164

minus332 minus1128 minus5128 minus13072 minus196 minus164 minus1256

minus21160 29128 minus35184 minus532 minus1256 minus 316

minus332 minus1512 minus8512 minus332

minus116 minus16

Figure 4The Feynman graphs contributing to the genus two free energyin the case of a single cut

14

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

The first two terms come from the dressing of the twist operators and the last term is the logarithm ofthe correlation function〈0|σ(aprime)σ(a)|0〉

The termF (2) in the genus expansion of the free energy is a sum of the contributions of allpossible genus two Feynman diagrams composed by the vertices and the propagators shown in Fig3The relevant diagrams are depicted in Fig4 The result is6

F (2) =1

micro21

(

minus 21

160

micro32

micro31

+29

128

micro2micro3

micro21

minus 35

384

micro4

micro1+

5

32

micro3

micro1dminus 49

256

micro22

micro21d

minus 105

512

micro2

d2micro1minus 175

1024

1

d3

)

+ micro harr microprime +1

micro1microprime

1

(

minus 1

64

micro2microprime

2

micro1microprime

1dminus 5

128

micro2

micro1d2minus 5

128

microprime

2

microprime

1d2minus 69

256

1

d3

)

(58)

The genus two free energy was first computed by Ambjornet al in [3] To compare with the resultof [3] we have to express the momentsmicron andmicroprime

n of the classical field in terms of the ACKM momentsMn andJn defined as

Mn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)n+12(z minus aprime)12=

1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicronminusk

Jn = minus 2

N

∮

dz

2πi

Jc(z)

(z minus a)12(z minus aprime)n+12= (minus1)nminus1 1

N

nminus1sum

k=0

(2k minus 1)

k(minus2d)kmicroprime

nminusk (59)

Substituting (59) in the answer found in [3] (note that some of the coefficients are corrected in [24])

N2F (2) = minus 181

480J21d

4minus 181

480M21d

4+

181J2480J3

1d3minus 181M2

480M31d

3+

3J264J2

1M1d3(510)

minus 3M2

64M21J1d

3minus 11J2

2

40J41d

2minus 11M2

2

40M41d

2+

43M3

192M31d

2(511)

+43J3

192J31d

2+

J2M2

64J21M

21d

2minus 5

16J1M1d4+

21J32

160J51d

minus 29J2J3128J4

1d+

35J4384J3

1d(512)

minus 21M32

160M51d

+29M2M3

128M41d

minus 35M4

384M31d

(513)

one reproduces the expression (58)

6 Discussion

The CFT formalism was developed before for the continuum limit of a class of matrix models whichreduce to Coulomb gas integrals similar to (11) such as the SOS and ADE matrix models [25]The classical solutions considered in [25] correspond to non-hyperelliptic Riemann surfaces with onehigher (or even infinite) order branch point at infinity and one simple branch point on the first sheetThe CFT formulation of these models provides a rigorous derivation of the Feynman rules for thegenus expansion obtained previously in [26 27]

In this paper we developed the CFT formalism for any classical solution of the hermitian one-matrix model We do not see conceptual difficulties to generalize it to any classical solution of theabove mentioned models including theO(n) model

The Feynman rules that follow from the CFT representation look similar to the diagram techniquedeveloped in a larger class of matrix models by Eynard and collaborators [4 28 29 30 31] If there

6 The author thanks A Alexandrov for pointing out a missing term in the unpublished extended version of [1]

15

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

exists an exact correspondence between the two formalismsthen the CFT formalism can be possiblyextended also for matrix models which do not reduce to Coulomb gas integrals It is likely that therecipe for transforming the Eynard formalism into an effective field theory proposed by Flumeet al[32] is the way to obtain this correspondence

AcknowledgmentsThe author thanks A Alexandrov B Eynard and N Orantin foruseful discussions

References[1] I Kostov ldquoConformal field theory techniques in random matrix modelsrdquoarXivhep-th9907060

[2] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanar DiagramsrdquoCommun Math Phys59 (1978) 35

[3] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoMatrix model calculations beyond the spherical limitrdquoNucl PhysB404(1993) 127ndash172 arXivhep-th9302014

[4] B Eynard ldquoTopological expansion for the 1-hermitian matrix model correlation functionsrdquoJHEP11 (2004) 031arXivhep-th0407261

[5] G Moore ldquoGeometry of the string equationsrdquoCommun Math Phys2 (1990) no 2 261ndash304

[6] AAlexandrov AMironov and AMorozov ldquoBGWM as Second Constituent of Complex Matrix ModelrdquoarXivhep-th09063305

[7] A S Alexandrov A Mironov A Morozov and P Putrov ldquoPartition Functions of Matrix Models as the First SpecialFunctions of String Theory II Kontsevich ModelrdquoarXiv08112825 [hep-th]

[8] AAlexandrov AMironov and AMorozov ldquoInstantons and Merons in Matrix ModelsrdquoPhysica D235(2007)126ndash167hep-th0608228

[9] A Marshakov A Mironov and A Morozov ldquoGeneralized matrix models as conformal field theories Discrete caserdquoPhys LettB265(1991) 99ndash107

[10] S Kharchev A Marshakov A Mironov A Morozov and S Pakuliak ldquoConformal matrix models as an alternativeto conventional multimatrix modelsrdquoNucl PhysB404(1993) 717ndash750 arXivhep-th9208044

[11] A Morozov ldquoMatrix models as integrable systemsrdquoarXivhep-th9502091

[12] G Bonnet F David and B Eynard ldquoBreakdown of universality in multi-cut matrix modelsrdquoJ PhysA33 (2000) 6739ndash6768 arXivcond-mat0003324

[13] A B Zamolodchikov ldquoConformal scalar field on the hyperelliptic curve and the critical Ashkin-Teller multipointcorrelation functionsrdquoNucl PhysB285(1987) 481ndash503

[14] L J Dixon D Friedan E J Martinec and S H Shenker ldquohe Conformal Field Theory of OrbifoldsrdquoNucl Phys1987(B282) 13ndash73

[15] V G Knizhnik ldquoAnalytic fields on Riemann surfacesrdquoPhys LettB180(1986) 247

[16] V G Knizhnik ldquoAnalytic Fields on Riemann Surfaces 2rdquo Commun Math Phys112(1987) 567ndash590

[17] V G Knizhnik ldquoMultiloop amplitudes in the theory of quantum strings and complex geometryrdquoSov Phys Usp32(1989) 945ndash971

[18] R Dijkgraaf A Sinkovics and M Temurhan ldquoMatrix Models and Gravitational CorrectionsrdquoAdvTheorMathPhys7 (2004) 1155ndash1176hep-th0211241

[19] T Miwa ldquoCLIFFORD OPERATORS AND RIEMANNrsquoS MONODROMY PROBLEMrdquo Publ Res Inst Math SciKyoto17 (1981) 665

[20] M R Gaberdiel A O Klemm and I Runkel ldquoMatrix model eigenvalue integrals and twist fields in the su(2)-WZWmodelrdquoJHEP0510(2005) 107hep-th0509040

[21] C Itzykson and J-B Zuber ldquoCombinatorics of the Modular Group II the Kontsevich integralsrdquoIntJModPhysA7(1992) 5661ndash5705hep-th9201001

[22] M Kontsevich ldquoIntersection theory on the moduli space of curves and the matrix Airy functionrdquoCommun Math Phys147(1992) 1ndash23

16

[23] M Bergere and B Eynard ldquoUniversal scaling limits of matrix models and (pq) Liouville gravityrdquoarXiv09090854 [math-ph]

[24] J Ambjorn L Chekhov C F Kristjansen and Y Makeenko ldquoerratumrdquoNucl Phys B449(1995) 681

[25] I Kostov and V Petkova ldquoNon-Rational 2D Quantum Gravity II Target Space CFTrdquoNuclPhysB769(2007)175ndash216hep-th0609020

[26] S Higuchi and I Kostov ldquoFeynman rules for string fieldtheories with discrete target spacerdquoPhys LettB357(1995) 62ndash70 arXivhep-th9506022

[27] I Kostov ldquoSolvable statistical models on a random latticerdquo Nucl Phys Proc Suppl45A (1996) 13ndash28arXivhep-th9509124

[28] L Chekhov B Eynard and N Orantin ldquoFree energy topological expansion for the 2-matrix modelrdquoJHEP12(2006) 053arXivmath-ph0603003

[29] L Chekhov and B Eynard ldquoMatrix eigenvalue model Feynman graph technique for all generardquoJHEP12 (2006)026arXivmath-ph0604014

[30] B Eynard and A P Ferrer ldquoTopological expansion of the chain of matricesrdquohttparxivorgabs08051368

[31] B Eynard and N Orantin ldquoAlgebraic methods in random matrices and enumerative geometryrdquoarXiv08113531 [math-ph]

[32] R Flume J Grossehelweg and A Klitz ldquoA Lagrangean formalism for Hermitean matrix modelsrdquoNuclPhysB812(2009) 322ndash342httparxivorgabs08053078

17