Linearizing W2,4 and WB2 algebras

ELSEVIER

23 March 1995

Physics Letters B 347 (1995) 260-268

Linearizing W2,4 and WB2 algebras S . Bellucci a~l, S. Krivonos a*b*2, A. Sorin a*b,3

a INFNJaboratori Nazionali di Frascati, PO. Box 13, I-00044 Frascati; Italy b Bogoliubov Theoretical Laboratory, JINR, Dubna, Russia

Received 26 November 1994 Editor: PV. Landshoff

PHYSICS LETTERS B

Abstract

It has recently been shown that the W3 and W, (*) algebras can be considered as subalgebras in some linear conformal algebras. In this paper we show that the nonlinear algebras W2.4 and WB2 as well as Zamolodchikov’s spin 5/Z superalgebra also can be embedded as subalgebras into some linear conformal algebras with a finite set of currents. These linear algebras give rise to new realizations of the nonlinear algebras which could be suitable in the construction of W-string theories.

1. Introduction

In the last few years a lot of attention has been devoted to nonlinear algebras. A great success was achieved in the construction of W-algebras from the iinear Kac-Moody ( affine j algebras by gauging some set of constraints imposed on the currents of the latter (see Refs. [ 1 ] and references therein for reviews). In this way Miura-like free-fields realizations [2] of W-algebras were also obtained. This deep connection between W- and afline algebras by no means implies that the currents of the former algebra can be constructed directly from the currents of the latter one [3,4]. ,This remarkable relation between linear and nonlinear algebras allows one to analyze the properties of W-algebras and the corresponding W-strings from the affme algebras point of view.

A different way to connect W-algebras to linear ones was proposed by two of us [ 51. We addressed

1 E-mail: bellucciQ1nf.infn.k 2E-mail: [email protected]. 3 E-mail: [email protected].

the question, whether linear conformal Lie algebras with a finite number of currents exist, such that they contain nonlinear W-algebras as subalgebras in some nonlinear basis? It has been proved, for the simplest case of W3 and W:*) algebras, that such linear algebras exist, have the structure of an extension of the ‘W- currents by some set of currents forming a realization of the given W-algebra.

In an effort to show that the existence of the linear structure is a general feature of the W-type algebras, rather than an exception valid only for the simple cases mentioned above, and motivated by the ex- pectation that the properties of the theories based on nonlinear algebras can be understood in the simplest way from the corresponding linear structures, we extend in the present paper the list of nonlinear algebras which admit linearization, We analyze from this point of view three W-type algebras: W2,4 algebra [ 61, the simplest fermionic nonlinear WB2 algebra [7], and Zamolodchikov spin 5/2 algebra [ 81 which exists only for some particular value of the central charge. We find that in all cases the linear conformal algebras

0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00002-X

S. Bellucci et al. /Physics Letters B 347 (199.5) 260-268 261

indeed exist and are connected with the aforementioned types of W-realizations, and contain the considered W-algebras as subalgebras. Using the linear conformal algebras obtained we find, as a first application, a new field realization of W2,4 algebra. At the level of the linear algebras, the intrinsic relation between the classical W2,4 and WB2 algebras is more transparent, in the sense that the linear algebra for classical W2.4 is simply a subalgebra of the one for WB2.

2. WB2 and W2,4 algebras

In this Section we fix our notation by displaying the explicit structure of WB2 [ 71 (or, using a different notation, W( 2, $, 4) ) and W2,4 [ 61 algebras and their relations in both the quantum and the classical cases.

The WB2 algebra is the simplest example of a superalgebra with a quadratic nonlinearity. Besides the standard Virasoro stress-tensor T(z) it contains two additional currents: the bosonic current U( z ) with spin 4 and the fermionic current Q (z ) with spin 5/2. This algebra exists for the generic central charge c and their currents obey the following operator product ex- pansions (OPEs) 4 :

T(Zl)7-(z2) c/2 2T T’

=zp2+ig+z3

T(zl)Q(zz) =sQP I !& G2 212

4u U’ T(z1)7J(z2) = 3 + E 9

2c/5 Q(zl)Q(zd = -

c2

(2.1)

(2.2)

(2.3)

+ [b,T” + b2U + bsL4] $9 (2.4)

U(z1)Q(z2) = zfi d’Q+%+[d3Q”+4(TQ)] i

+ [&Q”’ + &tT’Q) + &tT Q’)] $9

4 As usual, we wiIl write OPEs in the quantum as well as in the classical case, keeping in mind that for the classical case only one pair contractions took place. The currents in the r.h.s. of the OPES are evaluated at the point zz, and 212 = z1 - ~2.

WZl)WZ2) c/4 2T T’

=z,sz+T+T 212 212

+ [a$ + &&4 + ugT”] i

+ [$a, U’ + ;a2 L; + u4T”‘] $

+ [a#” + U6L[I + u7T””

1 + a&61 + a9L62 + @Oh3 + al&641 2

212

+ [al2U”’ + alsLy + u~4T”“’ + $ug L&

+ $a9 L&2 + $10 G3 + gll G,] $9 (2.5)

where the composites L4 (spin 4)) and L61 - L,54 (all with spin 6) are defined by

L4 = (TT),

‘561 = (T ~541, L62 = (T’T’) ,

L63 = (T U), L64 = (Q’Q) (2.6)

with normal ordering understood for the products of currents. The values of all coefficients in the quantum and classical cases are given in Table 1.

The W2.4 algebra contains only the bosonic currents {T(z),U(z)} withspins {2,4}. Thecurrent U(z) is primary with respect to the Virasoro stress-tensor T. Therefore the only OPE we need, in order to specify this algebra, is U( zl)U(z2). This OPE has the same form as (2.5) with ali = 0 and all other coefficients as given in Table 1.

We would like to stress that for both WB2 and W2,4 algebras we use the nonstandard definition of composite currents (2.6)) which are non-primary with respect to the corresponding stress-tensor T( z ) , in contrast with the papers [ 6,7] where primary composites have been used. Despite the less evident structure of the coefficients in the OPEs (2.4)) (2.5)) the use of non- primary composites gives a significant speed up of all calculations (at least twice as fast) which have been done with the help of the Mathematics [9] package OPEdefs [ lo].

Let us summarize some peculiarities of these alge bras.

First of all, let us notice that in the quantum WB2

algebra the coefficient b2 = ,/w before the

262

Table 1

S. Bellucci et al. /Physics Letters B 347 (1995) 260-268

Expressions for the coefficients in the OPES for WB2 and W2.4 algebras.

Coeff. Quantum WB2 Classical WB2 Quantum w2,4 ch%icd w2.4

6

al

a2

a3

a4

a5

ag

a7

08

as

a10

a11

a2

al3

a14

3&R4

42

I_,, 2

2c-29 -Kg-

5c+64 -R 2(c+2qd/6 4 176~+117c-2528

2(2c-1)(7c+68)R2

lti - 1662 -1645&U% 12(2c-1)(7c+68)Rz 24(72&13)

(2c-l)(7c+68)R2

-3(192-78ti-2368) 2(2c-1)(7c+68)R2

$$R4

0

c--4 R4 2(c+24)d6 3(13c2-131c-342)_ 2(2c-1)(7c+68)R2

ld -2832 -898cfS296 80(2+1)(7c+68)R2

3522 -704c-9125 2R1 R2R3

~OC? -462~? --2862c+9865 lWRzR3

108(32c-5) RlR2R3

-3(38~? -1669c-4930) 2RlRzR3

RIRS

RIRzR~ 2&J -1196c?+2953c+44150

8OR1 R2R3

z!p 2

27 R2

3 m

2fi

g

bl

h

b3

HerewesetRt=2~+25, R~=5c+22, Rg=14c+13, R4= (c+24)(c2 - 172c+ 196)

(5c+22)(7c+68)(2c- 1) ’

S. Bellucci et al. /Physics Letters B 347 (1995) 260-268 263

spin 4 current U(z) in the r.h.s. of (2.4) vanishes, for the particular value c = -g of the central charge. This means that for this value of the central charge the currents T( z ) and Q (z ) form (modulo null-fields) the closed quantum algebra which is the spin $ algebra of Zamolodchikov [ 81.

Secondly, we would like to stress that for the classical algebras all coefficients in the OPE U( zt ) U( ~2) for WB2 and W2,4 coincide (with the obvious exception of at 1 which vanishes for W2,4). This means that the classical W2,4 algebra can be obtained from WB2 by truncation, namely by dropping from OPEs the current Q ( z ) and all composites constructed from Q ( z ) . In other words, in the classical case the composite current (Q’Q) transforms homogeneously through it- self and Q ( z ) , and so it can be consistently dropped, together with Q( z ). In the quantum case the situa- tion changes drastically. As shown in Table 1, the values of the coefficients at - at4 for WB2 and W&4 are quite different and there exists no truncation proce dure connecting these algebras. The reason that sin- gles out the quantum case is the following: despite the existence of a primary spin 6 current which looks like (Q’Q) +corrections, its quantum OPE with U( z ) necessarily contains a term proportional to the U( z ) current in the right-hand side. So, in the quantum WB;! algebra we cannot exclude from the OPEs the current Q ( z ) and all composites constructed from Q ( z ) , without contradicting the Jacobi identities. That is why we need to consider the quantum W2,4 algebra inde- pendently from the WBp one.

In the next Section we construct the linear algebra which contains the W2,4 one as a subalgebra.

3. Linearization of the W2,4 algebra

In this Section we show that there exists a linear algebra Wi,t with a finite number of current which contains W2,4 as a subalgebra.

The main idea of our approach [5] is to extend the set of currents of the given nonlinear algebra by adding some new currents in such a way that: - the resulting set of currents still form a closed al-

gebra, - the extended algebra contains the Virasoro subal-

gebra and all other currents could be chosen to be primary with respect to Virasoro stress-tensor,

- there exists an invertible nonlinear transformation to another basis for the currents where all OPEs become linear.

In other words, here we would like to demonstrate that there is some linear; conformal algebra which contains the W2,4 algebra as a subalgebra in a certain nonlinear basis.

It is well known [ 3,4] that the currents of the nonlinear algebra obtained through Drinfeld-Sokolov re- duction, can be expressed in terms of the corresponding WZNW sigma-model currents. The novelties of our approach are as follows: - the number of currents in the linear algebra is the

same as in the extended nonlinear one and the transformation from one set of currents to another is invertible,

- the linear algebra is conformal, i.e. it contains the Virasoro subalgebra and all other currents are primary with respect to it. Let us demonstrate that for the W2,4 algebra the

corresponding linear algebra with the aforementioned properties does really exist.

The starting point of our construction is the follow- . . mg linear algebra Wi,;:

c/2 2T T’ 7YZl)T(Z2) = zfi + 2122 + E 3

T(z1)J(z2) = 2 + $ + & 9

W 2 T(z1)Q(z2) = --g + z12 9

2q1 Ul J(Zl)Ul(Z2) = - 3

212

J(Zl)J(Z2) = i 3 Ul (~1) Ul(z2) = regular,

(3.1)

where the central charges c and cl are connected with the U( 1) charge q1 as follows:

c = _ 12Oq: - 864; + 15

Cz ’

&-6& --G---’

(3.2)

Let us remark that we demand that the stress-tensor of the nonlinear W2,4 algebra coincides with the stress-

264 S. Bellucci et al. /Physics Letters B 347 (1995) 260-268

tensor of the linear algebra ( 3.1) . That is why we need to have the central charge cl in the OPE T( ~1) .I( ~2). Of course, we are free to pass by the transformation T + T + aJ’ to another basis where all currents are primary. However, in the basis under consideration all expressions look more tractable.

Now it is a matter of straightforward calculations to show that after the following invertible nonlinear transformation to the new basis (J, T, U), where

U = Ul + zl(T T) + zz(T’J)

+ zg(T J’) + ZA(T J J) + ~9”’ + zg(J J J J) + ZT( J’J J> + zg( J”J) + z9( J’J’) + zwJ” , (3.3)

the currents T and U form the W2,4 algebra (2.1)) (2.3)) (2.5) with the central charge c parametrized by 41 (3.2), provided the coefficients { ZI-210) are chosen to be

Zl = (8 - 274:) (3 - 44) al0

84( 1 - 8q;) ’

z2 = ( 1 - 4qT) (6004; - 4524: + 75) a*0

168q1(1- 8q;) I

z3 = ( 1 - 33 wwi - 4524; + 75) al0

84q1( 1 - 8q;‘) 3

75 z4=

6004: - 45247 +

84(8a: - 1) alo,

144Oqf - 186Oq; + 8654: - z5 =

1814; + 15

168qy<84: - 1) alo,

ze+

z7 = (3q: - 1) (6OOq;1- 4524: + 75) alo

42q1u - 84) ,

28 = (324; - 16s: + 3) (6OOqf - 452qf + 75) alo

336q;( 1 - 8q;) 9

z9 _ (484; - 364; + 7) (6OOq; - 452d + 75) alo

672q:C 1 - 84) ,

z,. = (96q~-68q;‘+19~-3)(600&45~+75) alo 20=x:( 1 -%;‘I (3.4)

and alo is defined as in Table 1:

8(7c - 115) 6 al0 = 2cf25 (5c+22)(14c+ 13) *

To obtain the classical limit, we need to make the following renormalizations of the U( 1) current J, central charge cl, and U( 1) charge 91:

J--t 15, fi

Cl -+&Cl, 41 -t&41,

(3.5)

and consider, as usual, the values of all coefficients in the limit

c+oo or 4140. (3.6)

The classical linear algebra W,,4 lin has the following form5 :

Z/2 ~(Zlu-(Z2) = --$- + 2 + g 9

7(Zl)C7(Z2) = -2iF/&iS

$2 +L+C 2122 212 ’

7(Zl)Ul(Z2) = 4241 u:

ZL+G?

2idELil %7(ZlM(Z2) = 9

212

L7(Zl)J(Z2) = 2 3

U1(z1>241(z2) =regula. (3.7)

The classical expression for the current U, forming together with I the classical W2.4 algebra, reads

(3.8)

5 We will use calligraphic and tilded letters, to distinguish the classical currents and charges from the corresponding quantum ones.

S. Bellucci et al. /Physics Letters B 347 (1995) 26&268 265

So, we have explicitly shown that the W$! algebra (3.1) ( VV$ (3.7) ) contains the quantum (classical) W2,4 algebra as its subalgebra in the nonlinear basis.

We postpone the discussion of the consequences of the linearization property to Section 5. In the next Section we show how to construct the linear algebra for the WB2 superalgebra.

4. Linearization of WB2 algebra

In this Section we explicitly demonstrate that WB2 is a subalgebra of a very simple linear conformal superalgebra.

The main novelties of WB2 algebra are, in compar- ison with W2,4, - the fermionic nature of the spin 5/2 current Q (z ) - the appearance of the spin 4 current U( z ) in the

OPE of two spin 5/2 fermionic currents. The last property means that the whole structure of WB;? algebra is encoded in the structure of the “super- symmetry” current Q (z ) . Moreover, the linear algebra WBP we are going to construct must reproduce at c = - 3 the linear algebra for the Zamolodchikov’s spin 5/2 nonlinear algebra. In addition, we know that the classical W2,4 algebra is a truncation of the classical WB2. In the case of the corresponding linear algebras, due to the absence of nonlinear composites, the YVgip, (3.7) must be a bosonic subalgebra of WBP.

Without going into details, let us show that WBP contains the bosonic {T, J, Ul} and fermionic {S, Ql} currents 6 , obeying the following OPEs 7 :

T(zl)T(zd c/2 2T T’

=zp,+,,2,+2 T(z1)J(z2) = 2 + ; + 5,

s/2 s T(zl)f3(~2) = z + G 3

T(z~)Ql(zd _ 5Qd2 I e: z:2 212 ’

6 We hope that the use of the same notation for the bosonic currents as in the previous Section does not give rise to confusion as to which algebra we are dealing with.

7For the same reason of simplicity as in the previous Section we work in the non-primary basis.

T(z1)& (~2) 4u1 u;

=z:,+Y?

42121 J(z1)Q1(z2) = 212 9

242Ul J(Zl)Ul(Z2) = - 9

212

S(Zl)RZ2) = i 9

Ql <zl)Ql (zd = - 212 ’

(4.1)

where the central charges c and c2 are parametrized by the U( 1) charge 42

c = _ 12Oq; - 125d + 30

29$ ’

c2 =2-4%)

(4.2)

and the coefficient b2 defined in Table 1 reads

b;-_ 6( 14c+ 13)

5c+22 .

In order to prove this, we do the following invertible transformation to the new basis {T, J, S, Q, U), where

Q = QI + ZIS” + ZP(J S’) + zs(J’S)

+.a(JJS) +zs(TS) (4.3)

and U is defined from the OPE in Eqs. (2.4))

2c/5 2T T’ Qh)Q(zd = x + 2132 + 2122

+ [blT”+bzU+bdTT)] A.

If the parameters {zl-zg} are chosen to be


(4.4)

and the central charge c is connected with q as in (4.2)) then the currents T, Q and U just form the WB2 algebra (2.5).

Thus, one concludes that WBP indeed contains WB2 as a subalgebra.

Let us close this Section with some comments. First of all, we would like to remark that the lin-

ear WBP algebra (4.1) and the coefficients {zt -zg} (4.4) do not contain singularities when c --+ -g

(q -+ fl) . Therefore, we can immediately read off the linear algebra for the Zamolodchikov’s spin 5/2 algebra with c = -2

-13/28 T(Zl)T(ZZ) = -

zfi

T(Zl)J(Z2) = 4@

$2 +J+&,

z:2

s/2 S’ T(Zl)S(ZZ) = -& + z 9

T(zt)Ql(z2) = = + g z:2 212 ’

J(zl>Ql(z2) = OQ1 , 212

f3(Zl)S(Z2) = i 3

J(Zl)J(ZZ) = 2 9

Ql (~1 > Ql(z2) = regul~. (4.5)

The corresponding expression for the spin S/2 current Qz, which together with the stress-tensor T forms the Zamolodchikov’s spin 5/2 algebra, reads as follows:

QZ = Ql + $j& St’- $$(JS') -&J'S)

(4.6)

One can immediately see that the bosonic subalgebra of WBP coincides with WF4 (3.7). While the classical W2,4 algebra is a truncation of WB2, the corresponding linear algebra W& is a simple subalgebra of WB!$‘.

Thus we showed that the linear superalgebra (4.5) In the next Section we consider some examples that contains the Zamolodchikov’s spin 5/2 as a subalge- can illustrate the use of the linearized algebra for both bra. W2,4 and WB2 algebras.

Secondly, as pointed out in the Section 2, the classical WB2 algebra contains the classical YV2,4 algebra, in the truncation limit Q -+ 0. It can be easily shown that the analogous relation exists also for the linear algebra WBP. In the classical limit c --+ co, after the redefinitions

J-+-J, S+ h

-I_& G

c2 -+ fi’c2 3 42 + &CT2 9 (4.7)

the classical linear algebra WB!f’ has the following form:

E/2 7(ZlV(Z2) = zfi + 2 + g 9

T(Zl )J-(z2) = -2iElv%

$2

+L+zL z:2 212 ’

s/2 S’ am = --& + & 9

7(Zl>&l(Z2) _ =1/2 I 8:)

2122 a2

7(ZlY41(22) = 2 + 2 3

idG !Zl c7(21)&1(22) = ---&-- 3

2iJi5Ul L7(ZlM(Z2) = 9

212

S(z1 V(z2) = g 9

L7(Zl)J-(ZZ) = $3

Ql(Z1)81(Z2) = 2&pUl

. 212 (4.8)

S. Bellucci et al. /Physics Letters B 347 (1995) 260-268 261

5. Conclusions and discussion

In this paper we extend the list of nonlinear algebras which admit linearization, by explicitly constructing the linear conformal algebras, with a finite set of currents, which contain W2,4 and WB2 algebras as subalgebras in a nonlinear basis. Thus, now we know that the first representatives of both the WA,, (W3 [ 51) and the WB, (WB2) series of nonlinear algebras can be linearized. Despite the lack, up to now, of a general algorithmic procedure to construct the linear conformal algebra for any given nonlinear algebra, it makes sense to conjecture that the possibility of linearization is a general property of nonlinear algebras rather than a peculiarity of some exceptional cases.

Let us finish by discussing some common properties of the linearization and the linear algebras.

First of all, we would like to note that, in order to linearize a nonlinear algebra, we need to extend it by adding some currents (J( z ) in the case of W2,4 and { J( z ) , S( z )} in the case of WB2 algebra). Thus, the first open question is as to which additional properties of the system are associated with these symmetries.

Secondly, it is interesting that the constructed linear algebras are homogeneous with respect to some currents (VI(Z) and{Ur(z),Qr(z)}inthecaseofW2,4 and WB2, respectively). This property means that we could consistently put these currents equal to zero and be left with the realization of the algebras in terms of an arbitrary stress-tensor and a spin 1 current J( z ) for W2,4, and a spin 1 current J( z, ) and a spin l/2 current S( z ) for the WB2.

Finally, let us remark that, owing to the invertible relation between the currents of the nonlinear and the linear algebras, every realization of Wz% or WB!f’ is a realization of W2,4 or WB2, respectively. So, the problem of constructing the realizations of these algebras is reduced to the problem of constructing realizations of the linear algebras W$ and WBP. In the rest of this Section we will present an example of such realization for the case of W2,4 algebra.

From the simple structure of the W$! algebra ( 3.1) it is clear that its most general realization includes at least two SC&V fields +i (i = 1,2) with OPEs

4i(zl)+j(z2) = -&jln(zl2) (5.1)

and, commuting with them, a Virasoro stress-tensor i;

having a nonzero central charge which we will denote as cr. Representing the bosonic primary current Ul ( z ) in the standard way by an exponential of 4i, we find the following expressions:

J = i4: ,

ul =sexP (ij/s&2 +2iql$l) ,

cr_3 (4-N+12#_ (2&-1)2 -( N-4qf 4:

> (5.2)

where N runs over non-negative natural numbers and s is an arbitrary parameter (due to a Ul resealing in- variance of the OPEs (3.1) s can be chosen to be 0 or 1).

In the case of s = 0 the field C#Q can be absorbed by the stress-tensor and we end with the following 42 independent realization:

T=Fo_ &)‘_ i(l --13qf)&‘,

J=ic${,

u, =o,

% = 1 _ wq: - II2 4: (5.3)

After substituting eqs. (5.3) into (3.3), we get the known realization of W2,4 algebra [ 111, whereas the use of eqs. (5.2) in (3.3) yields a new realization of W2,4 algebra, which may play a role for W2,4 string theory [ll].

A last comment concerns the realization (5.3)) for which the W& algebra is reduced to a direct product of a U( 1) algebra and the Virasoro one with the central charge CT0 (5.3). Then, the minimal Virasoro models [ 121 which correspond to the central charge

give rise to the following induced central charges (3.2):


c”z; = -2 (5(214 - 6d (3~2~) - 54 (2Ph *

(5.5)

They coincide with the central charges of W2,4 minimal models [ 131

c min = _ 2 (5D - 62) (3P - 5q”) p”4

(5.6)

for even values of p”.

Acknowledgement

We are grateful to L. Bonora, A. Honecker, K. Hom- feck, E. Ivanov, W. Nahm, V. Ogievetsky and S. Sciuto for many useful discussions.

Two of us (S.K. and A.S.) wish to thank INFN for financial support and the Laboratori Nazionali di Frascati for hospitality during the course of this work.

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Linearizing W2,4 and WB2 algebras

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