414

L

F

os as

richever and

ovi ov (

) have introduced a generalization of the Virasoro algebra which describes conformaltransformations on a Riemann surface of arbitrary genus.

e examine their genus one algebra. find explicitexpressions for its center . and discuss connections and symmetries among the algebra constants. We thenpr

_ to construct explicitly the corresponding superconformal algebra on the supertorus . An intriguingfeature of the

S odd spin structure algebra is that it is enriched by a central element which is anticommuting.

he,e the indices are halfintegers and the constantsen., Xnm are specified below .

e generalize this algebra to the supertorus . Ourmost striking algebra is the odd spin structure NS al-gebra :

1 .I a outlining

r done in collaboration withezinc

u and R.

epomechiel .

y coverage hereis eclectic. and I would refer readers to our paper fora

ore detailed discussion.

richever and Novikov(

)$ have pioneer

a framework for describing conformal transformations on

iemann surfaces of arbi-trary genus. of relevance in string physics and criticalsystems . Their generalization of the Virascro algebrato genus one . the torus. is:

3/1-

0[en , em] _

~men+m-a

a 6/2

3"c- _

da+1nm ~n+m-a-1 +-xnm ,

(9n

IC CENTER I THE SUP RCO FOR

L ALGEBRA O

THE SUPERTORUS

Z C OS

Energy Physics Division .

[en , em] =

a=-a/2

brk supported by the U.S .

rgonne National Laboratory. Argonne. IL

39. USA

aenmen+m-a +c Xnm,

1/2

, gm) _ - 2

va

amen+~.-a

s/2

,.- 26

iaa+1

cnm9n+m-a-1 + 2

onm ,

0920-5632/89/$03.50 © Elsevier Science Publishers H.V .(North-Holland Physics Publishing Division)

Nuclear Physics B (Proc. Suppl.) 11 (1959) 414-424North-Holland, Amsterdam

a[en , qm] _

Un,ngn+m-aa=-s/a

s/Z-I-

c26

hn�.a+1en+,~-a-1 - 4ô Pnma=_s/a

(2)The structure constants and cocycles are relatedamong themselves and are specified below. This alge-bra bears some correspondence to the Neveu-Schwarzalgebra . Nevertheless. observe that the fermionic mod-ular parameter b of the supertorus (which may bethought of as a gravitino zero mode) allows new op-erator terms on the right-hand-side of this algebra, aswell as the above fermionic center . parameterized bythe numbers Pnmm "

Since functions on the torus are doubly-periodic.they are elliptic functions$ . I will sketch below the con-venient elliptic function basis introduced by KN: howit specifies the conformal algebra : and hove we gener-alize both this basis and hence the KN algebra to thesupertorus . For questions not pertaining to eq.(2) .you may further consult other relevant work6 .

2.The torus has two periods . 2w1 and 2ws . with

Im(w2/wl) > 0. It may be defined by identifying thepoints on the complex plane x ;z z+2rnw1 +2nwa. forarbitrary integers rra and rc . Henceforth refer all pointsof the plane to this fundamental cell . Without loss ofgenerality, choose 2w1 = 1, 2w2 = r.

epartment of Energy. Division of High Energy Physics . Contract W-31-109-ENG-38 .

Meromorphic functions on the torus are doubly pe-riodic and are called elliptic functions' . An elliptic func-tion without poles has to be a constant : the order ofelliptic functions (number. of poles in a cell) is two orhigher : the sum of the locations of the poles of an ellip-tic function equals the sum of the locations of the zeros(equal in number) . properly weighted by their multiplic-ities : significantly. the residues of an elliptic function atits poles in each cell sum to zeros .

The prototype elliptic function is the WeierstrassP(z) function . with a double pole at the origin. Weier-strass further defined the odd quasi-elliptic functionss(z) (not the Riemann zeta function) and a(z) throughc'(z) = -P(z) and al(z)/or(z) = S(z) : S has a simple

Note that the constants

Im z

anr

(-1)n-1 asn(zo((2

-1)zo).

(4)

a

((

2n

+ 1)zo)

are an artefact of the lopsided normalization which sim-plifies behavior at zo at the expense of behavior at -zo.You may check that the zeros and poles of this ba-sis satisfy the abovementioned summation relations .Generic negative and positive n's have their poles atzo and -zo, respectively. while Ai/z . which obeys thegeneric formula . has none. and A_glz has to straddlethese two domains with one pole at each point . sinceone cannot have an elliptic function with a simple pole.This minor asymmetry is reflected in the orthonormality

C.K. Zachos/Ferrnionic center in superconformal algebra

2w2

2w1

2((ol + w2)

0, Re z

415

pole with residue 1 at the origin. while a(z) - z forz - 0. The functions P, S. and a stand in correspon-dence to the simply periodic functions -cosec=(z) . cot .and sin . respectively. They may be expressed in termsof the less handy but more standardized Jacobi thetafunctions.

In analogy with the genus 0 case . Krichever andNovikovl distinguish two points on the surface. (punc-tures) . which can be mapped to z = zo and z = -zo.They then provide a basis for all elliptic functionswith poles only at z = ±zo. Such functions may beexpanded in the complete basis of elliptic functions{A.(z)) . with n half-integer :

relations :

~~ 2;An(z)Am(z)

&+mo,

n,m 9& - I

dz-A

2i[A-~ (z))2 _ 4S(zo) - ZS(2zo) .The integrations are performed along a counterclock-wise path about the point zo . To prove these relations.use the fact that the sum of the residues of an ellipticfunction at its poles in any cell is zero . As a result . theintegrals may be evaluated alternatively through clock-wise integration about -zo . In the generic case . the

BASIS FUNCTIONS z ti zo z N -Zo

An(`z)

V~-$ o on+__ (z-zo z-F-2nzo 2zo 1e+ nz+z Q((2n~-1)zo) ~ ~ -2

(Z n-1n-16-ZO) tL~(Z ~ ~~_n-

A1f2 = 1 î 1

-zlc) a_ 2zo-

c2(!)aa .l, . z-Z, oa ZO)

( _ )_p-(

)-1

416

regular at either z = zo or z =rats vanish . except forn+m=

You may thus use the above conditions to evaluate

e expansion coefficients o an

elliptic function

itpoles a ±zo in terms of the An.

r example. sincethe derivative of an elliptic function is also an ellipticfunction. it t

may

e

an

in the same basis:

product A.(z)A,.(x) is. an

hence the inte

that

here

an hence

O

(1) is defined below ((18a)).

e(Z) =

Pyn,-

(Zo)A (Z) .

(6)

`ynm(

) =0,

In +

[ > 1,

(,) :A (-112,-3/2) ;

,7n,1-n() = n- 2 '

(gb)

11n,n(zo) =(

+

)D(1) (2zo(n + 1)

+ (n -

)D(1) (2z°(2 - n))

%,-1_n(zo) = (n +

) S' (2zo(n + 2)) - C'(2zo), ,(gd)

(8e)7-1 .12,-3/2(AID) = -a-3/2(ZO) .

The above then provides a matrix representation forthe derivative operator in this basis. The raising-preserving-lowering operation on the index of the

Enzn+l

Ln s zn+1a8z

zn+1

'..M+1 = ( - n)zn+m+1

[Ln , Lm ] = (

- n)Ln+

C.K. Zachos/Ferrnionic center in superconformal algebra

operand may e symbolized byThis may e used successively in su sequent manip-ulations which involve derivations. in conjunction withmultiplication of basis functions An . which also closesin the same basis (An}.

3.Consider no a conformal transformation z -"

z+ e(z), here z is a local complex coordinate . Thecorresponding generator LE defines a vector field:

LESc(z)±

9

and satisfies the (centerless) conformal algebra

ICRICHEV R-NOVIICOVE(z) =

e.An(Z)ncz+ ;

[en s em] =

LElLE' ] LE38sE9 '

where e1 a. ez - el r3sCS - (r3se1)es . One can treat anti-analytic transformations z -~z + ?(i) analogously : toavoid repetition . focus on the analytic ones .

Recall how this reduces in the genus 0 case . Thecoordinate z on the complex plane serves as a globalcoordinate on the sphere . by stereographic projection .Distinguish the points (punctures) at z = 0and z = oo.which lie in the infinite past and in the infinite future .respectively. Consider functions e(z) which are analyticfor z 96 0 , : such functions can be expanded in thebasis (zn+1). for n integer. These are the globally-defined functions on the doubly-punctured sphere . Inlight of the above basis, generalization to the torus fol-lows naturally. I proceed to compare functions. basesfor conformal transformation generators, their commu-tation structure, and centers for the sphere and thetorus with punctures at fzo. respectively :

3/zAn(Z) a. A.(Z)

c'm(Zo)An+m-s(Z)

cnm(ZO)en+m-e

(10)

The above

expressions hold for the generic caseInk ;6 â , Iml :0 j" , n + m 96 -2 .

For special valuesof n and m (e.g ., Inj = â , Imp _

, etc.) . they aremodified . T(z) is an analytic energy-momentum tensorof a conformal field theory on the respective 2-d mani-fold . whose moments . as indicated . provide an alterna-tive representation of the algebra : but now a non-trivialcenter is also specified . through the singularities of thestandard short-distance expansion for zl close to z2 .

T(zl)T(z2) ^'

c/2

+

2

T(z2)(zl - z2)4(z1-z2)2

1+

a2T(z2) + (finite),(zl - z2)

with c a positivé real number characterizing the partic-ular two-dimensional quantum field theory. The coun-terclockwise paths of integration run along equal-timesurfaces about the point zo . Standard manipulationsrelate equal-time commutators to this operator prod-uct :

< [en , e n] >=

L. =-0dz zn+1T(z)2ni

_1 j dz (zn+l)erzm+l12 27ri

J

dZ2Am(z2)

2~riAn(zl) < T(zl)T(z2) > .ZO

,(13)

The structure constants of KN may be evaluateddirectly from orthonormality :

o) _

dz [A,(z)aAn(z)]

~A-n-m+s(z),ZO 21ri

In(,Iml,In+m-s1 :0 1 ; n±m 96 -2 .(14)

In the generic case . they vanish for 1,91 > 3/2 and are2:

cn'm=m -n,

cn,,n =(n- m)D(l) (2za(n + m -1))

+ (m - 1/2)D(l) (2zo(m + 1/2))

- (n - 1/2)D(l) (2zo(n + 1/2)),

C.K. Zachos/Fermionic center in superconformal algebra

(12)

(1s)

en - -

dzA.(z)T(z)so 2vi

417

Xnm =-I j dZ12 j 2

$A!."(z)A.(z)

(11)

where 1)1 1, is defined in (18a) . Note the contrast

tVirasoro structure constants : here they depend ori zoand the modular parameter r through their dependenceon the functions or and C. Likewise [2] .

Xn,2-n =3.2 (n - 2)(n- 2){(n + 2)D11, (2zD(n + 2))

12 (n - 2){2 (n+ 2)(n + 2)D(2) (2zD(n + 2))

+ 2(n - 2)(n - 2)D(2) (2zo(2

- n))

+ (n + I)(n- 3)D(s) (Zzo(n + I))2 2

2

x D(1)(2zo(2 - n))),

Xn,-n =

+ (n - 5)DO) (2xoj2- n))),

(17c)

12{ô(n + 2)(n + 3)(n + 2)D(s) (2zo(n + I))+ 1(n-6

2

21)(n - 3) (n - 2) D(s) (2zo(2 - n))

+ I(n + 2)(n + 3)(n - 2)2

2x D(2) (2zo(n + 2I)) DM (2zo(2n)

+ 1(n- 1)(n - 3)(n +1)

2 2 2 2x D(2) (2zo(2 - it)) D(l) (2zo(n + 2)) )

(17d)

(17e)

Xnm(zo) = 2 2~An(z)A.(z) , (16)ZO

which we compute:

Xn,�e =®, n+m>3; (17a)

Xn,s-n = -12(n 2)(n 2) (n 5)' (17b)

418

In this notation . the antisymmetry of the cocycleXn,m = -Xm,n is manifest. It is evident in eq. (16)that the cocycle vanishes when either index is 1/2.

b have given the values of Xn,m only for n+m >- 0.and c.m for positive superscripts . However. for n+m <0. no further calculations are necessary. because of theidentity

-1X°n.°m(zo)

=an(-zo)am(- ZO) Xn,m(-ZO) s

which relates values of the cocycle for n + m < 0 tothe known values for n + > 0. This identity. as wellas the corresponding one for the structure constants

InI,ImI,In+

+sI 96 2 ; n+m 96 -2,(20)

follow from the reflection property of the basis :

A.(z, -zo) = A-n(z, zo) an(-zo)

,

InI ,96 2 .

(19)

(21)

To see this . consider the expression (16) for the cocy-çle. now written to display the parameter zo :

here C is a closed counterclockwise path around zo .Sending zo --> -zo yields

Xnrn(ZO) = 12 c 2~Ao"(z9 zo)A.(z, zo) ,

(22)

1 r dz ,,Xnm(-ZO) = 2 d? 27riA

(z, -zo)Am(z, -zo) , (23)

where

isa closed counterclockwise path around -zo.Since the integrand is elliptic and has poles only at fzo.and

y virtue of the fact that the sum of the residues

4m(ZO) =

jc 27ri [An(Zi zo) asAm(ze Zo

)JA_,-.+.(z,zo)

(25)Now send zo --+ -zo. to obtain

D 27r8 [An(Z2 -Z°)C9sAm(Z,-zo)J

A_n_,n+.(Z, -ZO)

= -an(-ZO)am(-zo)a_n-vn+.(-ZO)X

Consequently.

(zo)ce-n,-m =

-1

(27)an(-zo)am(-zo)a-n-m,-. (-zo)

crim (-zo) ,

for InI, Iml, In + m+ s) 96 â , n + m # -2: and, sincea_,n(zo) = -1/am,(zo) . the identity (20) results .

From the first identity. and from eq.(17a), it imme-diately follows that Xn,,n = 0 for n + m < -3. That

.Ic2[A-.(z, "~)cl.A-m(z,Zo)]An+m- .(Z,zo))

_ -an(-zo)am(-zo)a-n-m+.(-zo)c=;,-m(zo) .(26)

Xn,m = 0

for

In+mI > 3,

(28)

so that the KN algebra has a "local" central extension.in parallel with the vanishing of the structure constantsca.. for IsI > 3/2. You may visualize these directlyby successively applying (8) in the "trident" schematicintroduced above. For instance, triple application of (6)on (16) yields

_1 dzXnm

_-12 Jsu 27ri

A,,Ap -in,-,t 7k,-!`i!,-pk,l,p

(29)

C.K. 2achos/Fermionic

have introduc the notation

center in superconformal algebra

of an elliptic function at its poles in any cell is zero . it

(1)(2zom) --- S(2zo ) - mC(2zo) , (18a)follows by use of (21) that

(2)(2zom) - e(2zom) -me(2zo) Xnm(-zo)1 dz

=�,

-_ r

27riAn(z,-zo)A.(z, -zo)

+ [D(1)(2zom)]2 , (18b)12 Jc

(s)(2zo ) --_ S"(2 ) - mC"(2zo) - (-zo)a~

n(-zo) 12 2iA!-".(Z, zo)A-,.(z, zo)

+ 3 (_)(2zom)D(1)(2zom)c

(24)

- 2 [D(1) (2zom)]s . (18c)which is the identity (19) .

Likewise. for. InI, ImI, In+m-sI 0 j' ; n+m 96 -2 .

Evidently. the indices of the composite tridenttreespan 7 levels . In particular . triple application of (8b)instantly produces the leading term (17b) . Relationsof this type are summarized in the table at the end ofthis talk .

For the non-generic cases involving Al/2 and A-1/2whose behavior at z ,-- -zo is not canonical . eq.(21) ismodified and yields corresponding modified relations .

The identities (19) and (20) may also be used toexhibit an automorphism of the K algebra . Recall thatthe Virasoro algebra

[L� , L�,] = (m - n)L�+. + 12 (W- n)ô�+.,o (30)

has the automorphism L� -o, -L_� . c -+ -c . Simi-larly, the KN algebra has the automorphism

en --i, an(-ZO)e-.

ei -+ el,

e_1 --' -e_1,

C --* -C ,z 2 s

provided that also go - -zo in all functions ofzo such as the structure constants c".(zo) and thecocycle X..(zo) . From a�(-zo) = -an(zo) anda� (zo)a_� (zo) = -1. note that the square of the trans-formation (31) is 1 .

It is worth emphasizing that the characteristic formof the conformal algebra on a surface is predicated onthe global basis on that surface . as illustrated above.rather than on the short distance expansion (12) whichis universal .

For spinning (supersymmetric) strings and super-conformal critical systems. a relevant symmetry is theN = 1 superconformal symmetry. It is best describedin N = 1 superspace with local complex analytic coor-dinates (z,6) . where ® is an odd (Grassmann) coordi-nate obeying f2 = 0. An infinitesimal superconformaltransformation is given by

z -~ z + V + 2(DV)®,

® -~ e + 2I DV,

(32)

where

is an even infinitesimal super-analytic function . and thesuperspace covariant derivative D is

D

8o+ äz '

Da _äz

(34)

C.K. Zachos/Fermioni c center in superconformal algebra

(31)

V = V (z, ®) = e(z) + ®+P(z)

(33)

The corresponding generator ,Cv is

w =

z+2(DV)D ,

(35)

and obeys the (centerless) superconformal alg

here

[wl , 1:V2] = w$^va , (36)

VI A Vs = VI a. Vs + 2(DVI) (D 2) .

(37)

On the N = 1 supersphere. with global analcoordinates (z, ®) . we distinguish the two points (z =0, ®) and (z = oo, ®) : and we look for a suitable basisfor the functions e(z) and O(z). which are the even andodd components of V(z, ®) :

O(z) =

Orzr+a ,(38)r

where ifi, are Grassmann parameters . The index r cantake on either halfinteger (°'Weveu-Schwarz") or inte-ger ("Ramond") values. We use the bases in (38) toconstruct a superfield basis for the superconformal al-gebra . For V = zn+i . i.e. vanishing O(z) .

LV - Ln =z°+1- + -(n + 1)f- .(39a)

01Z ae

For V = ficzr+' (where ic is an odd parameter) . i .e .vanishing e(z) .

£v --> 2Gr

with

Gr = zr+;

a

a) .`r3® -®laz

(39b)In this basis . the superconformal algebra (36) reads

[L� , L,n ] = (m - n)Ln+,n ,

[L., Gr] = Cr - 2 ) Gn+r ,

{Gr , G.) = -2L,+.,

(40)

which is the familiar (centerless) Neveu-SchwarzRamond algebra .

The supertorus is the supersymmetric generaliza-tion of the torus. and is parameterized by superellipticfunctions4 . A spinor field O(z) on a bosonic torus canhave one of four possible spin structurese characteriz-ing its sign change under parallel transport around eachcycle:

420

z --> z+ 1z -~z+T INNIN

These conditions also foliopoints on the superplane

C.K. Zachos/Fermionic center in superconformal algebra

The first 3 structures are call

"even" . and the fourthspin structure (+.+) is called "odd" .

r the even spin structure (+.-) . the requirementthat V(z, 0) = E(z) + 1%(z) be invariant under paral-lel transport around both cycles implies that V(z, S)satisfy

V(Z, O)= V(z + 1, +0),

V(Z, 0) =V(Z + r, -0) .(41)

if

eidentify the following

(z, 0) ;e (z + 1, +0 ,(z,0) sks (z + r, -0) .42

In fact . (2) define a supertorus with (+.-) spinstructure . The definition for the o other even spinstructures. (-.+) and (-.-) is analogous .

supertorus with odd spin structure is defined byidentifying the points

(z+1,0) ,(z,0) m (z+T+06 , 0+b) , (43)

where 6 is a new odd modular parameter . A functionV(z, ®) on a supertorus with odd spin structure satis-fies

V(z,®) = V(z+1, 0) ,

V(Z, 0) = V(z+T+06 , 0+a) .(44)

superanalytic superfield which satisfies (41) or (44)is a "super-elliptic' function.

The number of spin structures doubles upon intro-ducing the o punctures : (z = ±z,0,0) . If the abovecycles flip their characteristic signs for fermion trans-port when pulled over either off the punctures. they arein the Ramond sector : for no change of sign . they comprise the

eveu-Schwarz sector. Here . I report on theodd spin structure

S sector. leaving the other. lessdramatic. ones to be examined in our paper . Our solu-tion to the problem offinding a suitable superfield basisfor functions with the superperiodicity (3) is the fol-lowing. Introduce a function O(z) . which satisfies

46(z + 1) = O(z) ,

0(z+ r) = O(z) - 1 .

(45)

a consequence. the quantity ® -I- O(z)b is super-elliptic . and it is therefore the desired generalization of

®.

e further demand that q$(z) be single-valued andthat its poles (if any) be at fzo . We choose

«z)

2ai (S(z - ze) - 2(z - zo)g1]

(46)

where %71 = C(1/2) . Our choice is somewhat asymmet-ric. as ßß(z) has a pole at zo and not at -zo. and leadsto some asymmetries in the expressions below .

Our basis for super-elliptic functions analytic atz0fzois

Â.(z,®) = A.(z) + q6(z)A.(z)98,

B.(z,0) = An(z) (0(z)b + 01 ,for n half-integer. The corresponding basis of super-conformal transformations is

A V 1 Aen = An

49Z-{- 2 (DA. D ,

-in AA,, = -AnclsA9n + Z(DÂ.) (DA,.)

An A bin --- -An51,Bm + 2(DAn)(DBn)

O(z) -

tai [P(z -zo) + 2till

_-iai1

[A-3/2(z) + P (2zo) +217j] .

(47)

(45)9n = -2 [b

.az49 - 2(DBn)D

For a mnemonic of the halfintegral indices of thefermionic generators gn . compare (39) with the O(ao)expansion of the above . namely en = Anâa +12A;,®â+O(6) and gn = An(é -$A-) + O(ô) .88

To specify the relevant superconformal algebra . weevaluate the components of (37) :

cnmAn+m-a + d E dntnlhn+m-a-18=-3/2

e=-6/2(49a)

Bnl~B�~ = BnÔsBm - 2(DB. (DB.)

1 1/a

s/z= -2 r fnmAn+m-a +Ô r han+n'Bn+m-a-1 9

8=-1/s

8=-s/a(49b)

a

AunmBn+m-a - Ô

ha+1nm An+m-e-1

e=-s/z

.=-3/2(49c)

where we introduce the new structure constants d;,n .h,", and u.". . detailed presently. Observe that

the derivative of O(z) is A-s/Z(z), up ¢o constants:

(50)

Now introduce the structure constants f.". :

where

1/2

a=-1/2

Further define the structure constants unm:

s/2

A.(z)A. (z) - 2A.(z)A�. (z) =

u.,.A.+.-.(z) .a=-s/2 3/2

112

w w(52)

=

UnmBn+m-a + ,6

f.O

g.Direct calculation of An A A. yields

a=-s/2

a=_1/2(61)

AnA Am = (Anc3:Am) + [0 (An~3,,Am)~I 06

For special values of n and m. these equations ares/2

$/2

slightly modified . As a consequence . (36) dictates'_

An+m-o + ®Ô

cnm ,̀r~lAn+m-a

the generators en and 9n obey the algebra (2) .cnam "

Now

where

=-s/2

im-s/2

As in the bosonic case. the center is derived from(53)

thesupercontour integral representation of Lv in termsof the super-energy-momentum superfield T(z, ®) .

OIAn-

27r=A-$/2An

27râ [P(2zo) +2171]An2

-r anAn-k , where the z integration is performed along an equal-

(54)

time counterclockwise path about the point zo.where we introduce the constants an:

usual. integration over the odd coordinate

is deft

ano _ 1

-27rs[p(2zo) +2171]

by

ar1

___ 1 ;-1/2~2_ 1 1/2

(55)27riJ_3/2 .n ,

n

27rs

3/2,nWe then find that the last term of eq.(53) is given by

dnm -cnm ~n+m+k-a ,k

Consequently (49a) follows . Next . evaluate

As above . the last term is

-06

d..A.+.-a ,

(56)a=-s/2

bn A Bm=-2{A�A.+O(A.A.)'ea} + O'(A�Am)eb1/2

1/1

2_--

fnmAn+m-a + ®Ô

,fom~~An+m-a2a=-1/2

a=-1/2

-®Ô

OL~yy,An+m-a ,a=-1/2

CrX Zachos/Fermionic center in superconformal algebra

421

-3/2 <- s < 7/2 .

(57)

(53)

(59)

nm -1 nm

n+m+k-a ,k

so that (49b) follows. Finally.

Ân A

.=(A�A;n - 2£.A,.)(® + 06) + ®(A.

_ _

dz"21ri

f

en = -

dzIdO.Â.(z,O)T(z,O),

.. 27r$

gn = 2

dz fd®Bn(z, ®)T(z, ®) ..0 27rs

6/4 3012

w w

T(z1, d,)T(z2, 02) - z2+ 2

z,22 T(z2, 02)

-1/2<s<5/ ;

®=1 ,

The projection on our basis An and 1n is then

' .Cv. -J~ 27rti f

d®2V2(z2, ®_) !~

f27râ

x Vl(Z1, ®1)T(Z1, ®1)T(z2, ®2) .

V(z,®)T(z, f) ,(62

(63)

Equal-time commutators are expressed in terms of op-erator products:

The operator product expansion for T(z1 , ®1)T(z2,02)for z1 close to z2 necessarily reduces to the operatorproduct expansion on the superplane

+ 1 12D2T(z2, ®2) + La2T(z2, O2) ,

(66)

422

here c _ lc. 0 12 _®1 - ®B. and z12 _ z1 - z2 - ®108 .

recast this result as follows:

T(z1901)7"1z2,02)

here S(®1 - ®a) = ® 1 - 02 . This

['CV., Zv']

j

dz f2ri

_c dz+

f8 .. 2wi

lenseml =

cnmen+m-e -

c dzd®

2 is. i7-ri f

and for [en, 9ml the center reads

- c

dz4 80 27ri

~D1b(01 - ®_)

4

(z1 - za)$36(01-02)2 (zi-z2 ,

+ !Dl

D2T(zs, 02)2

(zi - ab)

C.K. Zachos/Fermionic center in superconformal algebra

h may now specify .Cv, andr instance. consider the case

Recalling eq.(49a) and performinthesecond term of the right-hanobtain

which appears in eq.(2) above . given the definition (16) .

Similarly, for (gn, g n) . the center is

8 27ri n m

(70)

A

DA;; _ - ~bp. "

(71)

Upon performing the ® integrations . these centers re-duce to the expressions

ore specifically.

tknm(ZO)_

dz~n(z)Am(z)

(72)-".

2rri

Pnm(zo) _ ~ 2~rOB(z)[

An(z)Am(z) - An(x)A,n(z) ] "(73)

+(n+ 1)(n-1)D(1 ) (2zo(n + 2)I2 2

x D(1) (2zo(2 - n))) .

All centers are local . It bears repeating that the cen-ter of the commutator [en , g.] is proportional to ô . andthus fermionic. in some contradistinction to usual cen-ters of infinite algebras .

I summarize much of the above information andsome of the inter-relations among constants in the fol-lowing

implies= 0, In + ml > 2 ; (74a)

V2T(z, ®)

V1" .(68)

On,Z-n = (n - I)(n - 2) (74b)

.Cv, t rough egs.(64) .

k

AA(67)

On,l-n =(n - g-l ){(n + 2)D(1) (2zo(n + 1))V1 = A,. . Vs = elm .

+ (n - 3)D(1) (2zD(!

2 -n)) )

(74c)

g the ® integration ind-side of eq.(68) . we

fin,-n = {1(n+ 1)(n+ 3)D(2) (2zo(n + 2)/2 2 2

d®+1nm9n+m-s-i +2(n - 1)(n -

3)D(2)(2zo(2 - n»2 2 (74d)

dz(69)

Note that . by dint of the simplicity of the leadingterm in (8b) . the leading term of all of these constantsmay be found essentially by inspection from the above.e.g.

3/2 _

_n

_1unm - (m- 2 - 4/ ,

(75)-3/2 -_

anam

_n

_ _1

~nm

an+m+3 2 (2 -M

cf. eq.(20). except for special values of n and m. Sim-ilarly.

1/2 = 1

1/2 = ana.

76fnm an+m+1/2

except for special values of n and m. Likewise . theleading term in the cocycle of eq.(73) is

Pn.4-n = -

(n - 2)(n - 2)

(77)irs

5.Corresponding bases and algebras obtain for the re-

maining three NS spin structures and the four Ramondones. In the latter case. a branch cut is intboduced

C.K. Zachos/Fermionic center in superconformal algebra

between zo and -zo. to yield the requisite extra signchanges. and the fermions now have integer indices.Since bases for higher genera have been considerKAI (unpublished) and the last of ref.[5] . you might ex-pect the above supersymmetrization procedure to gothrough in these cases as well. Further desirable exten-sions of our work include the higher N superconformalalgebras . non-orientable surfaces. the investigation ofthe supermodular response features of our basis. andthe representation theory of the algebras involved (in-cluding its relation to that of the sphere).

The algebras presented here should provide aframework for an operator formalism for loop string the-ory [2]. and facilitate calculational techniques for correlation functions on the torus. They may also shedfurther light on the closed string interaction vertex .

by

I wish to thank the University of Miami and the Or-ganizers of this Conference for the pleasant and stimu-lating environment in which this work was initiated andpresented. respectively.

~ TERMS CONSTANT

Table

CONNECTING RELATIONS

3 -ynm = 2m nAm2 fn'm = Z,.An m~â-n-m+s7 ~nm lü= 12

11 2dzs8 n m - 12 EM in,-k^Ik,-l'il,m =12 Ekin,-1c0km

4 cnvn = I Z~~[Anaz m]A-n-m+s _ Ek'im,-kfnks+k-m

- 7n,-kfmks+k--n

4 unm = 2[An m2 nAm]A-n-m+s _Ek^fm,-kfn+ka-m 1a+k nk - Zin,-kfnk

5 "~~~Ynm = 2

da Alln m = Elt'in,-k^1k,m

3 dZ2~

O'nAk-n 2L[6k,o(P(2zo) + 21l1)

+bk,l.f3/2,n+6k,2f13/2,n]7 pnm --_ f Z~~4`(A" -m nAm) = Ek,1 at+m'Yn,-k7k,-1 - at+k,in,-k'im,-16 9

dn~n-k k

- Ekcsnm ~n+m+k-s

4s

hnms-k k

-Ek fnm ~n+m+k-a

424

1 . . inc u . .Ic

.

. h s.

. i

itta r an'~

' ( aa

ri e, 1® i® rin -

rane an

J.1

1: Js . .C . .

. . Visan

ver an

.

.

ovi

v.

nct,

al.

1.I. .

'

ver an

.P.

ovi ov.1 .

)

C. . Zachos/Fermionic censer in superconformal algebra

ie. and C. . a-

n . A C®arse ~nrid e niversi Press .

Chan rasek aran . dlipticrla . ) .

a in . Co

. ath.

s. 11a in . i n S

r

BCd

T'

ries.ias.

.

unstatter. R . .

annnat an

lenu .1 7) : .

Ra in and P.G.®. Freund. Cornna .

ath.Phys.11~(198 ) 131 .

.

.

onora,

.regola,P. Cotta-Rarnusino. and-artellini. Phys.Lett. ~®5 (1988) 53 : L .nora.

.artellini.

. Rinaldi, and J. Russo .P ys.Lett. ~®6

(1988)

: ®. Fairlie . P.Fletcher. and J . IVuyts.

urham preprint T'P88/ -

at .P ys ., in press :

. o ick. in prepara-tion: J . lberty. . T'aorenina, and P.

n ~a~l.Cornrn .

at .Phys., in press: L . ~onara.

. Lu~o,-atone . and J . Russo. SISS

preprint,

ay19

ei er and

.

itten.

ucLPhys. ~~~6 (1986)272.