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1995

LNF-95/048 (P)JINR E2-95-376

NULL FIELDS REALIZATIONS OF W3 FROM W (sl(4), sl(3)) AND

W (sl(3|1), sl(3)) ALGEBRAS

S. Belluccia∗, S. Krivonosb† and A. Sorinb‡

aINFN-Laboratori Nazionali di Frascati, P.O.Box 13 I-00044 Frascati, ItalybBogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia

Abstract

We consider the nonlinear algebras W (sl(4), sl(3)) and W (sl(3|1), sl(3)) and find their real-izations in terms of currents spanning conformal linearizing algebras. The specific structureof these algebras, allows us to construct realizations modulo null fields of the W3 algebra thatlies in the cosets W (sl(4), sl(3))/u(1) and W (sl(3|1), sl(3))/u(1). Such realizations exist forthe following values of the W3 algebra central charge: cW = −30,−40/7,−98/5,−2. Thefirst two values are listed for the first time, whereas for the remaining values we get the newrealizations in terms of an arbitrary stress tensor and u(1) × sl(2) affine currents.

September 1995

∗E-mail: bellucci@lnf.infn.it†E-mail: krivonos@thsun1.jinr.dubna.su‡E-mail: sorin@thsun1.jinr.dubna.su

1 Introduction

Nonlinear W algebras have been extensively considered from different points of view for thelast years (see e.g. [1]). Despite many impressive results the question of constructing real-izations is still open even for the simplest W3 algebra. The well known tensoring procedurefor constructing the new realizations starting from the known ones, which works perfectlyin the case of linear algebras, cannot be applied to W algebras due to their intrinsic nonlin-earity. That is why each new realization gives us a deeper understanding of the structure ofnonlinear algebras.

One of the possible ways to construct the realizations of W algebras is the conformallinearization procedure [2–4]. The main idea of this approach is to embed nonlinear W al-gebra as a subalgebra in some linear conformal algebra W lin. Once this is done, then eachrealization of the linear algebra W lin gives rise to a realization of W . Recent results in thisapproach [4,5] show that fortunately a wide class of nonlinear algebras admits conformal lin-earization and therefore the realizations of these algebras can be constructed systematically.

However the problem of constructing the realizations of nonlinear algebras is far from itscomplete solution, due to the existence of the so called realizations modulo null fields [6–11].The simplest example is given by the W3 algebra, where we allow for the spin 4 null operatorU4(z) in the operator product of two spin 3 generators W (z):1

W (z1)W (z2) =cW

3z612

+2T

z412

+T ′

z312

+[

2U4 +3

10T ′′ +

32

22 + 5cW

Λ]

1

z212

+

[

U ′4 +

1

15T ′′′ +

16

22 + 5cW

Λ′

]

1

z12, (1.1)

with Λ = (TT ) − 310

T ′′. For U4(z) to be a null operator we must require that there beno central term in the OPE of U4(z) with itself, so that < U4U4 >= 0. The OPE (1.1)together with the standard OPEs with the Virasoro stress tensor T (z) are not exactly theW3 OPEs. Nevertheless U4(z), being a null operator, can only generate null fields in itsOPEs. Therefore the set of null currents generated in a closed algebra is an ideal and canbe consistently set to zero leaving us with the realization of W3.

It is clear that the construction of the realizations modulo null fields is a rather compli-cated task and can be naturally divided in two parts.

Firstly, because the OPE (1.1) is a subset of the OPEs of some algebra W, larger thanW3, we need to know this algebra, together with the condition for its contraction to W3 (thatis, the spectrum of the central charge that makes the equation < U4U4 >= 0 satisfied). Afirst attempt to classify the possible W algebras which admit a contraction to WN is madein [11], where a conjecture for the spectrum of central charges corresponding to the givencontraction is proposed.

Secondly, we need to construct the realizations of the algebra W for the specific values ofthe central charge that allow the contraction of W to W3. Up to now, such realizations of W3

have been constructed for the following central charges: cW = −2,−114/7 [8], cW = 4/5 [7],cW = −10,−2, 4/5 [10], cW = −98/5,−10,−2, 3/4 [11], cW = −2, 4/5 [9].

The aim of this Letter is to construct new realizations of the W3 algebra modulo null fields

1The currents in the r.h.s. of the OPEs are evaluated at the point z2 and z12 = z1 − z2 with normalordering understood for the products of currents.

1

starting from the simplest nonlinear (super)algebras W (sl(4), sl(3)) and W (sl(3|1), sl(3))which possess the following properties:

• the algebras include the OPE (1.1),

• the algebras can be linearized.

The linearizing algebras are useful for the construction of the realizations, while the pres-ence of the OPE (1.1) guarantees the existence of the contraction to the W3 algebra. Newrealizations exist for the following values of the W3 central charge: cW = −40/7,−2,−30(W (sl(4), sl(3)))2, cW = −2,−98/5 (W (sl(3|1), sl(3))). We would like to stress here thateven for the previously known values of the central charge the constructed realizations of W3

modulo null fields are new.

2 W (sl(4), sl(3)) and W (sl(3|1), sl(3)) algebras

In this Section we present the explicit structure of W (sl(4), sl(3)) and W (sl(3|1), sl(3))algebras in the quantum case.

The W (sl(4), sl(3)) algebra can be constructed, as hinted by its very name, by consideringthe principal embedding of the sl(2) algebra into the sl(3) subalgebra of the sl(4) [1,12]. TheW (sl(4), sl(3)) algebra exists for a generic central charge c. Its currents obey the followingOPEs:

T (z1)T (z2) =2c

z412

+2T

z212

+T ′

z12, T (z1)U(z2) =

U

z212

+U ′

z12,

T (z1)G(z2) =2G

z212

+G′

z12, T (z1)G(z2) =

2G

z212

+G

′

z12, T (z1)W (z2) =

3W

z212

+W ′

z12,

U(z1)U(z2) = −c1/2

z212

, U(z1)G(z2) = −2G

z12, U(z1)G(z2) =

2G

z12,

G(z1)G(z2) =c2/3

z412

+a1U

z312

+[

1

3T + a2(UU) + a3U

′

]

1

z212

+[

W + a4(TU) + a5(U′U) +

1

6T ′ + a6U

′′ + a7(UUU)]

1

z12,

G(z1)W (z2) =b1G

z312

+ [b2(UG) + b3G′]

1

z212

+ [b4(TG) + b5(UUG) + b6(U′G) + b7(UG′) + b8G

′′]1

z12,

G(z1)W (z2) = −b1G

z312

+[

b2(UG) − b3G′] 1

z212

−[

b4(TG) + b5(UUG) − b6(U′G) − b7(UG

′) + b8G

′′] 1

z12

,

W (z1)W (z2) = A

{

cW

3z612

+2T

z412

+T ′

z312

+[

2U4 +3

10T ′′ +

32

22 + 5cW

Λ]

1

z212

2Only two of these values cW = −40/7,−2 coincide with the conjecture in [11].

2

+[

U ′4 +

1

15T ′′′ +

16

22 + 5cW

Λ′

]

1

z12

}

, (2.1)

where

T = T +1

c1

(UU), (2.2)

Λ = T T −3

10T ′′, (2.3)

U4 = d1(GG) + d2W′ + d3(WU) + d4(TT ) + d5(TUU) + d6(T

′U) + d7(TU ′)

+d8T′′ + d9(UUUU) + d10(U

′UU) + d11(U′U ′) + d12(U

′′U) + d13U′′′. (2.4)

and the values of all coefficients are given in the Table. We introduce the coefficient A inthe r.h.s. of the W (z1)W (z2) OPE in (2.1) for convenience. For sure, it can be set to one byrescaling the current W (z), at the cost, however, of further complicating the coefficients inthe Table.

Let us note that the spin 4 current U4(z) is defined to be primary with respect to thenew stress tensor T (z)3

T (z1)T (z2) =cW

2z412

+2T

z212

+T ′

z12

, T (z1)U4(z2) =4U4

z212

+U ′

4

z12

(2.5)

and both T and U4 have regular OPEs with the u(1) current U(z)

U(z1)T (z2) = U(z1)U4(z2) = regular. (2.6)

Thus, the currents T (z), W (z) and U4(z) belong to the coset W (sl(4), sl(3))/u(1).We would like also to stress that there is no possibility to redefine the currents of

W (sl(4), sl(3)), in order to avoid the appearance of the U4(z) current in the r.h.s. of theOPE W (z1)W (z2). Therefore, the W (sl(4), sl(3)) algebra does not contain the W3 one as asubalgebra.

The W (sl(3|1), sl(3)) superalgebra contains currents with the same conformal spins asthe W (sl(4), sl(3)) ones: the Virasoro stress tensor T (z), a bosonic spin 1 current U(z), adoublet of fermionic spin 2 currents G(z) and G(z), and a bosonic spin 3 current W (z).So the only differences in the contents of W (sl(3|1), sl(3)) and W (sl(4), sl(3)) algebras isthe statistic of the spin 2 currents doublet. This is why the OPEs for the W (sl(3|1), sl(3))algebra can be written in the form (2.1) with the same definitions for the composite currents(2.2)-(2.4) and the coefficients given in the Table.

3Due to the regular OPE of the spin 3 current W (z) with U(z), W (z) is still a primary current, also withrespect to T .

3

Coeff. W(sl(4),sl(3)) W(sl(3|1),sl(3)) Coeff. W(sl(4),sl(3)) W(sl(3|1),sl(3))

c c1(4c1+13)4(c1−8) − c1(2c1+7)

4(c1+8) b12(c1+2)(5c1−16)

9c21

2(c1−8)(c1+2)(5c1+16)9c2

1(c1+8)

c2c1(c1+1)(c1−8) − c1(c1+2)

2(c1+8) b22(5c1−16)

3c21

2(c1−8)(5c1+16)3c2

1(c1+8)

cW4(c1+1)(c1+2)

c1−8 −2(c1+2)2

c1+8 b3(c1+8)(5c1−16)

18c21

(c1−8)(5c1+16)18c2

1

a14(c1+1)3(c1−8) −2(c1+2)

3(c1+8) b44(c1−8)

3c1(c1+4) − 83c1

a23

c1−8 − 1c1+8 b5

4(5c1−16)3c2

1(c1+4)

8(c1−16)3c2

1(c1+8)

a32(c1+1)3(c1−8) − c1+2

3(c1+8) b62

c1+42

c1+8

a44

3c1

43c1

b72(c1−4)(c1+16)

3c21(c1+4)

2(c1−16)3c2

1

a53

c1−8 − 1c1+8 b8

c31+4c2

1+80c1−256

18c21(c1+4)

(c1−4)(c1+16)18c2

1

a62(c2

1−2c1+24)

9c1(c1−8) −c21+8c1+48

9c1(c1+8) A −5c1−1618c1

− (c1−8)(5c1+16)18c1(c1+8)

a74(11c1−16)9c2

1(c1−8)

− 4(c1−16)9c2

1(c1+8)

Coeff. W(sl(4),sl(3)) W(sl(3|1),sl(3))

d1 − 72c1

(5c1−16)(c1+4) − 144c1

(5c1+16)(c1−8)

d236c1

(5c1−16)(c1+4)72c1

(5c1+16)(c1−8)

d3288

(5c1−16)(c1+4)576

(5c1+16)(c1−8)

d4 − 12(c1−8)(11c1+20)(5c1−16)(c1+4)(10c2

1+41c1−68)

24(c1+8)(11c1+20)(5c1+16)(c1−8)(5c2

1+9c1−68)

d5 −12(91c

2

1+260c1−704)

(5c1−16)(c1+4)(10c21+41c1−68)

48(29c2

1−36c1−704)

(5c1+16)(c1−8)(5c21+9c1−68)

d6 = d748

(5c1−16)(c1+4)96

(5c1+16)(c1−8)

d836(c2

1−c1+4)

(5c1−16)(10c21+41c1−68)

36(c1+4)(c21+8)

(5c1+16)(c1−8)(5c21+9c1−68)

d94(1153c3

1+924c2

1−19584c1+25600)

(c1−8)c21(5c1−16)(c1+4)(10c2

1+41c1−68)

8(47c31+1156c2

1−1280c1−25600)

(5c1+16)c21(c1−8)(c1+8)(5c2

1+9c1−68)

d1048(11c1−16)

(c1−8)c1(5c1−16)(c1+4)−96(c1−16)

(5c1+16)c1(c1−8)(c1+8)

d1124(c1−2)(23c3

1+133c2

1+140c1+192)

(c1−8)c1(5c1−16)(c1+4)(10c21+41c1−68)

−24(7c4

1+2c3

1−220c2

1−560c1−768)

(5c1+16)c1(c1−8)(c1+8)(5c21+9c1−68)

d124(178c4

1+213c3

1+1476c2

1+17056c1−32256)

(c1−8)c1(5c1−16)(c1+4)(10c21+41c1−68)

−8(31c4

1+251c3

1+1004c2

1−4832c1−32256)

(5c1+16)c1(c1−8)(c1+8)(5c21+9c1−68)

d134(c2

1−5c1+48)

(5c1−16)(c1−8)(c1+4)4(c2

1+14c1+96)

(5c1+16)(c1−8)(c1+8)

Table.

3 Contractions of W (sl(3|1), sl(3)) and W (sl(4), sl(3)) to

W3 algebra

As stated in the Introduction, our purpose is building new realizations of W3 modulo nullfields and finding the corresponding values of the central charge. From the explicit OPEsof the W (sl(4), sl(3)) and W (sl(3|1), sl(3)) algebras (2.1) we can see that the currents T (z)(2.2) and W (z) form a W3 algebra, modulo the spin 4 current U4(z) which is present in ther.h.s. of OPE W (z1)W (z1). This spin 4 current U4 in both cases is expressed in terms ofthe basic currents (2.4). Therefore we can require that the current U4(z) be a null operator,i.e. < U4U4 >= 0. For this, all we need to ask is the vanishing of the central term in theU4(z1)U4(z2) OPE. The corresponding equation is satisfied only for some special values ofthe central charge.

4

Next, we give the results for both W (sl(4), sl(3)) and W (sl(3|1), sl(3)).

3.1 W (sl(4), sl(3)) case

Here we list for completeness the vacuum expectation values for the currents T , W, U4 con-tained in the coset W (sl(4), sl(3))/u(1)

< T T > =2(c1 + 1)(c1 + 2)

(c1 − 8), (3.1)

< WW > = −2(c1 + 1)(c1 + 2)(5c1 − 16)

27c1(c1 − 8), (3.2)

< U4U4 > =576(c1 − 4)(c1 + 1)(c1 + 2)(c1 + 6)(2c1 − 1)

(c1 − 8)(c1 + 4)(5c1 − 16)(10c21 + 41c1 − 68)

. (3.3)

Thus for the values c1 = −6, 12, 4 which correspond to the following values of the W3 central

charge cW :

c1 = −6 ⇒ cW = −40

7, (3.4)

c1 =1

2⇒ cW = −2, (3.5)

c1 = 4 ⇒ cW = −30, (3.6)

the spin 4 current U4 becomes a null operator. All other poles and zeros of the vacuumexpectation value < U4U4 > provide us with further contractions of the algebra, where thespin 3 current W (z) and even the stress tensor T (z) become null operators.

Let us note that only the first two values of the W3 central charge cW = −40/7,−2 followfrom the conjecture of [11]. Therefore the spectrum of central charges for the contraction ofthe W (sl(4), sl(3)) algebra to W3 proposed in [11] is not exhaustive.

3.2 W (sl(3|1), sl(3)) case

The same calculations for the W (sl(3|1), sl(3)) superalgebra give the following results:

< T T > =−(c1 + 2)2

(c1 + 8), (3.7)

< WW > =(c1 − 8)(c1 + 2)2(5c1 + 16)

27c1(c1 + 8)2, (3.8)

< U4U4 > = −576(c1 − 12)(c1 − 1)(c1 + 2)2(c1 + 4)2

(c1 − 8)(c1 + 8)2(5c1 + 16)(5c21 + 9c1 − 68)

. (3.9)

So, the current U4 is a null operator for the following values:

c1 = −4 ⇒ cW = −2, (3.10)

c1 = 1 ⇒ cW = −2, (3.11)

c1 = 12 ⇒ cW = −98

5. (3.12)

5

This concludes the determination of the central charges spectrum for both W (sl(4), sl(3))and W (sl(3|1), sl(3)), when these algebras are contracted to the W3 one. In order to completeour task, we need to construct the realization of W3 modulo null fields explicitly. In the nextSection we will show that a straightforward way to construct such realizations comes fromthe conformal linearization procedure [2–4] applied to the algebras under consideration.

4 Null fields realizations of W3 algebra

One of the most important questions when considering nonlinear W algebras is the construc-tion of their realizations in terms of free fields or affine currents. The lack of a tensoringprocedure for constructing new realizations starting from the known ones makes the task offinding new realizations of the nonlinear algebras rather difficult. Moreover, in many casesit is unclear which set of currents we need to use, in order to construct such realizations.

A solution to this problem was proposed in [2–4]. The idea of this approach is to embedthe given nonlinear algebra into a conformal linear one which contains the former nonlinearalgebra as a subalgebra. Of course, such linearizing algebra contains more currents, incomparison with the nonlinear one. However, after performing such a linearization, thequestion of constructing realizations becomes almost trivial, because any realization of thelinear algebra gives rise to a realization of the corresponding nonlinear one.

4.1 Linearization of W (sl(4), sl(3)) algebra

Fortunately, for W (sl(4), sl(3)) such a linear algebra which contains it as a subalgebra isknown [4]. It contains in the primary basis, besides the Virasoro stress tensor T (z), fourcurrents with spin 1, i.e. U(z),J+(z),J3(z),J−(z) forming the u(1) × sl(2) affine algebra,and two additional currents G+

1 (z) and G+2 (z) with the unusual spin 3K+4

2K, where K is the

level of the u(1) affine algebra.4 The complete list of OPEs for this linear algebra reads asfollows:

T (z1)T (z2) =(3 − 2K)(3K − 4)

2Kz412

+2T

z212

+T ′

z12,

T (z1)U(z2) =U

z212

+U ′

z12, T (z1)J+(z2) =

J+

z212

+J+

′

z12,

T (z1)J−(z2) =J−

z212

+J−

′

z12

, T (z1)J3(z2) =J3

z212

+J3

′

z12

,

T (z1)G+1 (z2) =

(3K + 4)G+1

2Kz212

+G+

1

′

z12

, T (z1)G+2 (z2) =

(3K + 4)G+2

2Kz212

+G+

2

′

z12

,

U(z1)U(z2) =K

z212

, U(z1)G+1 (z2) =

G+1

z12

, U(z1)G+2 (z2) =

G+2

z12

,

J+(z1)J−(z2) =2 − K

z212

−2J3

z12

, J+(z1)J3(z2) = −J+

z12

, J+(z1)G+1 (z2) = −

G+2

z12

,

4The currents of the sl(2) affine algebra are related to those in the paper [4] as follows: J3 = J11 ,J+ =

J21 ,J− = −J1

2 .

6

J−(z1)J3(z2) =J−

z12, J−(z1)G

+2 (z2) =

G+1

z12, J3(z1)J3(z2) =

K − 2

2z212

,

J3(z1)G+1 (z2) = −

G+1

2z12, J3(z1)G

+2 (z2) =

G+2

2z12. (4.1)

One may wonder how it could be possible to construct the currents of W (sl(4), sl(3)) fromthe currents of the linear algebra (4.1) which possess completely different spins. The answerbecomes clear after defining the new stress tensor T (z)

T (z) = T + J ′3 −

K − 2

KU ′ . (4.2)

With respect to this stress tensor the currents of the linear algebra (4.1) have the followingspins:

Currents U J+ J− J3 G+1 G+

2

Spins 1 0 2 1 3 2

So, with respect to the stress tensor T (z) (4.2) the currents of the linear algebra (4.1) possessthe spins needed, in order to construct from them the currents forming the W (sl(4), sl(3))algebra. Moreover, after imposing the following relation between the free parameter c1 inW (sl(4), sl(3)) and the affine level K in the linear algebra:

c1 = 8 − 6K , (4.3)

the central charge for the stress tensor T (z) (4.2) coincides with the central charge of theVirasoro subalgebra of (2.1). Therefore we can identify these two stress tensors (that is whywe used the same letter in the definition (4.2)).

Now it is a matter of straightforward calculations to find the expressions for the remainingcurrents of the nonlinear W (sl(4), sl(3)) algebra in terms of the currents spanning the linearalgebra (4.1)

U = U + 2J3, G = J−,

G = G+2 −

(T J+)

3(K − 2)−

(J+J+J−)

3K(K − 2)+

2(J+J3J3)

3K(K − 2)−

(J+J′3)

3(K − 2)+

2(UJ+J3)

3K(K − 2)+

(UUJ+)

2K(K − 2)−

2(UJ ′+)

3K−

2(J ′+J3)

3K+

(1 − K)(U ′J+)

3K(K − 2)+

(3 − 3K + K2)J ′′+

3K(K − 2),

W = G+1 −

(J+J′−)

3K+

4(5K − 4)(J3J3J3)

9(K − 2)(3K − 4)2+

2(UJ+J−)

3K(K − 2)−

8(K2 − 5K + 4)(UJ3J3)

3K(K − 2)(3K − 4)2+

(5K − 4)(UUJ3)

3(K − 2)(3K − 4)2+

(12 − 11K)(UUU)

9K(3K − 4)2+

(4 − K)(UJ ′3)

3K(3K − 4)+

2(1 − K)(J ′+J−)

3K(K − 2)+

4(K2 − 5K + 4)(J ′3J3)

3K(K − 2)(3K − 4)−

K(U ′J3)

3(K − 2)(3K − 4)+

(5K − 4)(U ′U)

6K(3K − 4)+ +

2(T U)

3(3K − 4)

(3K3 − 22K2 + 68K − 48)J ′′3

18K(K − 2)(3K − 4)−

(3K + 2)U ′′

18(3K − 4)−

2K(T J3)

3(K − 2)(3K − 4)−

T ′

6. (4.4)

7

The expressions above, in spite of their rather complicated appearence, allow us to con-struct the realizations of the nonlinear W (sl(4), sl(3)) algebra starting from any given re-alization of the linear algebra (4.1). The simplest realization corresponds to the case whenboth the G+

1 and G+2 currents are vanishing (i.e. they are null fields in the algebra (4.1)) and

so we are left with the realizations of the nonlinear W (sl(4), sl(3)) algebra in terms of anarbitrary stress tensor T (z), and the u(1) × sl(2) affine currents U(z),J+(z),J−(z),J3(z).

Let us note that the three exceptional points K = 0, 4/3, 2, where the transformations(4.4) become singular, correspond to the central charges c1 = 8, 0,−4 in the W (sl(4), sl(3))algebra. At these points some of the coefficients in the Table are also singular and hencethe currents that span the W (sl(4), sl(3)) algebra must be redefined, in order to avoid thesingularities (see e.g. [8]). So, the appearance of singular terms in the transformations fromthe linearizing algebra (4.1) to the nonlinear W (sl(4), sl(3)) one is dictated by the structurerelations for the W (sl(4), sl(3)) algebra we start with.

Thus, we succeed in the construction of the realizations (4.4) for W (sl(4), sl(3)). For thefollowing values of the parameter: K = 7/3, 5/4, 2/3, which correspond to the central charge(3.4)-(3.6), the realization (4.4) is the realization of W3 modulo null fields.

4.2 Linearization of W (sl(3|1), sl(3)) superalgebra

The conformal linearizing superalgebra for W (sl(3|1), sl(3)) has not been constructed sofar. However the bosonic case we considered in the previous Section gives us some hintshow such linear conformal superalgebra can be constructed. It is natural to assume thatthe linearizing superalgebra for W (sl(3|1), sl(3)) must contain the same spins as its bosoniccounterpart (4.1) with some currents being now fermionic ones.

Without further justifications, we write down the linearizing superalgebra which containsthe following currents: a Virasoro stress tensor T (z), four currents with spin 1, i.e. thebosonic currents U(z),J3(z) and the fermionic ones J+(z),J−(z), as well as two additionalfermionic currents G+

1 (z) and G+2 (z) with spin 3K−32

2(K−4). The currents obey the OPEs

T (z1)T (z2) =(K + 11)(3K + 8)

10(K − 4)z412

+2T

z212

+T ′

z12, T (z1)U(z2) =

U

z212

+U ′

z12,

T (z1)J+(z2) =J+

z212

+J+

′

z12, T (z1)J−(z2) =

J−

z212

+J−

′

z12, T (z1)J3(z2) =

J3

z212

+J3

′

z12,

T (z1)G+1 (z2) =

(3K − 32)G+1

2(K − 4)z212

+G+

1

′

z12, T (z1)G

+2 (z2) =

(3K − 32)G+2

2(K − 4)z212

+G+

2

′

z12,

U(z1)U(z2) =K

z212

, U(z1)G+1 (z2) =

G+1

z12, U(z1)G

+2 (z2) =

3G+2

z12,

U(z1)J+(z2) =2J+

z12, U(z1)J−(z2) = −

2J−

z12, U(z1)J3(z2) = −

K − 4

5z212

,

J+(z1)J−(z2) = −K − 4

10z212

+J3

z12, J+(z1)G

+1 (z2) = −

G+2

z12,

J−(z1)G+2 (z2) =

G+1

z12, J3(z1)G

+1 (z2) = −

G+1

z12, J3(z1)G

+2 (z2) =

G+2

z12, (4.5)

8

where the free parameter c1 in W (sl(3|1), sl(3)) and the affine level K in the linearizingsuperalgebra (4.5) are related as follows:

c1 =8 + 3K

5. (4.6)

Analogously to the bosonic case, we can express the W (sl(3|1), sl(3)) currents in termsof the currents of the linear superalgebra (4.5):

T = T +2(K + 1)

K − 4J ′

3 +1

2U ′,

U = U + J3, G = J−,

G = G+2 +

10(T J+)

3(K − 4)−

75(J+J3J3)

(K − 4)2−

10(J+J′3)

3(K − 4)2+

50(UJ+J3)

(K − 4)2+

25(UUJ+)

3(K − 4)2−

10(K + 6)(UJ ′+)

3(K − 4)2−

20(K + 11)(J ′+J3)

3(K − 4)2−

5(K + 6))(U ′J+)

(K − 4)2+

(K2 + 7K + 56)J ′′+

3(K − 4)2,

W = G+1 −

100(J+J−J3)

(K − 4)2+

10(J+J′−)

3(K − 4)+

100(61K2 + 327K + 416)(J3J3J3)

9(K − 4)2(3K + 8)2+

40(K + 1)(T J3)

3(K − 4)(3K + 8)+

10(T U)

3(3K + 8)−

100(UJ+J−)

3(K − 4)2+

125(K + 4)(11K + 16)(UJ3J3)

3(K − 4)2(3K + 8)2+

250K(K + 6)(UUJ3)

3(K − 4)2(3K + 8)2+

25(K + 16)(UUU)

9(K − 4)(3K + 8)2−

5(7K + 32)(UJ ′3)

6(K − 4)(3K + 8)+

20(K + 6)(J ′+J−)

3(K − 4)2−

5(31K2 + 32K − 224)(J ′3J3)

6(K − 4)2(3K + 8)−

5(7K2 + 64K + 32)(U ′J3)

6(K − 4)2(3K + 8)−

5(K + 16)(U ′U)

6(K − 4)(3K + 8)−

T ′

6+

(K + 1)(3K − 52)J ′′3

9(K − 4)(3K + 8)+

(3K2 − 4K + 368)U ′′

36(K − 4)(3K + 8). (4.7)

The formulas (4.7) give us the desired realizations of the W (sl(3|1), sl(3)) superalgebrain terms of the currents of the linear superalgebra (4.5). For the values K = −28/3,−1, 52/3this realization is the realization of W3 modulo null fields.

5 Conclusion and outlook

In the present Letter we construct explicitly the nonlinear algebras W (sl(4), sl(3)) andW (sl(3|1), sl(3)) and find their realizations in terms of currents spanning the correspondinglinearizing conformal algebras. The specific structure of these algebras allows us to constructmodulo null fields realizations of the W3 algebra that lies in the cosets W (sl(4), sl(3))/u(1)and W (sl(3|1), sl(3))/u(1). Such realizations exist for the following values of the W3 algebracentral charge: cW = −30,−40/7,−98/5,−2. The first two values are listed for the firsttime, whereas for the last two values we get the new realizations in terms of an arbitrarystress tensor and u(1) × sl(2) affine currents.

Let us finish by presenting a conjecture about the spectrum of central charges for theminimal models of the considered W (sl(4), sl(3)) and W (sl(3|1), sl(3)) algebras. The ideacan be described as follows: it is evident that for both linear algebras (4.1) and (4.5), after

9

putting the null currents G+1 and G+

2 to zero, one can construct the stress tensor Td(z) whichcommutes with the remaining currents. The central charge cd of this stress tensor Td(z) isstill connected with the central charges of the nonlinear W (sl(4), sl(3)) and W (sl(3|1), sl(3))algebras and the latter can be expressed in terms of cd. This is the general situation for theconformal linearization procedure [2–4]. Moreover, in all known cases of linearizing algebras,the minimal models for the Virasoro algebra spanned by Td(z) reproduce the minimal modelsfor the nonlinear algebras. One can assume that the same is true for the W (sl(4), sl(3)) andW (sl(3|1), sl(3)) algebras. If this conjecture is correct, then the central charge cd for theVirasoro minimal models

cd = 1 − 6(p − q)2

pq

will induce the following values for the central charge of the minimal models:

c =(8p − 15q)(4q − 3p)

pqfor W (sl(4), sl(3)) ,

and

c =(8p − 3q)(2q − 3p)

pqfor W (sl(3|1), sl(3)) .

Of course, this conjecture must be checked by the standard methods.An interesting extension of the results presented here comes from the consideration of the

nonabelian case, which corresponds to the algebras W (sl(N+3), sl(3)) and W (sl(3|N), sl(3)).For these algebras, the same procedure for constructing the realizations of W3 modulo nullfields will give the series of the central charges (with a manifest dependence on N). Thiswork is in progress.

AcknowledgmentsWe are grateful to E. Bergshoeff, E. Ivanov, V. Ogievetsky and M. Vasiliev for many

useful discussions. Two of us (S.K. and A.S.) acknowledge the hospitality at INFN-LNFwhen this work was undertaken. S.B. wishes to thank JINR and especially Rita and VictorOgievetsky for their kind hospitality when this work was completed.

This investigation has been supported in part by a grant of the INFN, Fondo ScambiInternazionali, the Russian Foundation of Fundamental Research, grant 93-02-03821, andthe International Science Foundation, grant M9T300 and INTAS, grant 93-127.

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