Singular vectors of the Virasoro algebra

Physics Letters B 273 ( 1991 North-Holland

) 56-62

PHYSICS

LETTERS

Camhrrdge

CR3 9EU:

B

Singular vectors of the Virasoro algebra Adrian Departrnrnt

Received

Kent (!f,4pplied

17 September

A4athrmntics

and Theoretical

I99 1; revised manuscript

Ph.wcs.

received

Univers~t~~ qf Carntvidge.

S11wr Sirwt.

C’K

I October I99 I

We give expressions for the singular vectors in the highest weight representations of the Virasoro algebra. We verify that the expressions - which take the form of a product of operators applied to the highest weight vector - do indeed define singular vectors. These results explain the patterns of embeddings amongst Virasoro algebra highest weight representations.

Conformal field theory relies on a description of the Virasoro algebra’s highest weight representations, and in particular on the classifications of the levels at which representations have singular vectors [ 1,2] and the embedding relations amongst these vectors [3,4]. These embedding relations are encoded in the irreducible Virasoro characters [ 5 1, from which the partition functions of conformal field theories are built [6-S]. While this information is enough for most conforma1 field theoretic purposes, there are several applications [9-l I ] in which explicit expressions for the singular vectors are needed. The first relevant work was the beautiful paper of Malikov, Feigin and Fuchs [ 121, which gives expressions for the general singular vectors in finite dimensional Lie algebra Verma modules. Feigin and Fuchs [4] have also presented some partial results describing projections, and asymptotic properties, of the Virasoro algebra singular vectors. Benoit and Saint-Aubin [ 131 (BSA) found remarkable explicit expressions for the subclass of the singular vectors L;,_~ in which either p or q is 1. Recently, Bauer et al. [ 14,151 have rewritten the BSA expressions in a compact form in which their singularity is manifest. and in which very interesting connections to integrable systems and to W-algebra theory appear. Bauer et al. have also given a new algorithm by which any vector z+,,~can in principle be calculated: we shall discuss this later. In this letter, we give expressions for all the singular vectors u,~:,,.~. show how these expressions explain the embeddings 56

0370.2693/91/$

of the Virasoro algebra’s highest weight representations, and sketch proofs of these results. First let us recall some basic facts and describe the results of Benoit and Saint-Aubin. The Virasoro algebra has commutation relations [L,,,, -&,I = (fn-

[L,,, Cl

n)L,,,+,, + &

=o

(tn’-tfl)&,,-,,C

, (1)

The Verma module P’(h, c) is the irreducible representation which contains a vector ) h) such that L,,,)h)=O,

iftH>O,

L,lh)=hlh), Clh)

=clh)

,

(2)

and which has a basis comprising the states L_ ,,... L_,,lh) with i,&...>i,.>O. We have the decomposition V(h.c)=

@ V,,(h.c), ,z=O.I.Z.

(3)

where the level n space b’,/,,( h, c) is the eigenspace of Lo with eigenvalue h + n. Define a singular vector in V(h, c) to be a vector L’,lying in some V,,(h, c) for n> 0, with the property that L,,,r=O,

ift?z>O.

(4)

It is known [ l-41 that there is a singular vector at level N in V(h, c) if and only if, for some positive integers p and q and complex number t, we have N=pq, and 03.50 0 1991 Elsevier Science Publishers

B.V. All rights reserved.

Volume 273, number 1,2

PHYSICS LETTERS B

c=c(t)=13-6t-6t

~

[(L

12 December 1991

,)O.L,,,] = Z ( I e I - ( m + 2 - r ) ( a + l - r ) ) ~l=l k r = l

h=h:,.+( t ) =~(p2-1)l-½(pq-1)+](q=-l)t

'.

(5)

It is also known that the singular vector at level N. when it exists, is unique up to scalar multiplication. Thus, given p and q, the singular vector v.:,., is a function of t. In fact, one can write z':,.,(t)= O:,.q(t) Ih,,,~:(t) ~, where

O,,.qCt)=

Y~ a:/qCt)L 1,

(6)

Itl-t+q

and the a}"q(t) depend polynomially on t and I ~. Here the sum is over sequences I = {i~..... i,,} of positive integers ordered so that i~>...>i,,, we write L t = L ,,...L ,,, and l l l = i j + . . . + i , , , and take the coefficient of ( L , ) ~ ' q to be 1. However, this turns out not to be the most convenient form in which to describe the operators Or. ~(t). In the cases when p = 1 or q= 1. BSA obtained remarkably simple expressions for the operators:

O,,.,(t)=

Z

cvCi, ..... i , , ) ( - t )

"L ,,

t = ', i+ ....,in i

I11--P

o,.~,(t) =

cuCi, ..... i , , ) ( - t ) " L _ l .

y.

(7)

1= i il ,...,in I F/I q

.....

/,,)=

[][

\(r-k).

(8)

l<~k
k4 ~]=l!J tbr any x

Our first step in generalising these results follows the ideas of Malikov-Feigin-Fuchs [ 12 ] by extending the enveloping algebra of the Virasoro algebra to include operators of the form (L_~)" for arbitrary complex values ofa. Thus, as well as the relations ( 1 ), we have

[ L .... ( L - l ) a ] - -

if m < 0 ,

(9cont'd)

and

(L ,)"L , = L

,(L

l ) a ~ - - - ( L _ l ) a+r

( L , ) " ( L ,)~'=(L_l)~'+".

(10)

(That is, we are considering the central extension of the algebra generated by differential operators z"d and generalised pseudodifferential operators d". ) Denote the algebra generated by L,,,, ( a n d the (L_I)~ by I7". Define the Vrepresentation I~(h, c) to be a generalised Verma module with a vacuum vector ]h~ and on which C acts as the scalar c. That is, eqs. (2) hold and the vectors L ,,,...L ..... ( L _ t ) " l h ) , with n,>...>~n,.>2 and a unrestricted, form a basis for f"( h, c). Note that ( L _ , ) " l h ) is not zero, even when a is negative, so that ph ) is neither highest nor lowest weight in f"(h, c). We shall be interested in the Virasoro singular vectors in f'(h, c) - that is, those vectors z' lying in some Lo eigenspace and obeying cq. (4). We say an infinite sum in V is a well-defined operator if it has the form

ao(L_~)"+ ~ a l L _ I ( L _ , ) "

ill

(11)

1

These sums are over all sequences of positive integers summing to p or q, without any ordering restriction. The coefficients are defined by c,(i,

X L ...... (L. l) a "

r

'"+~ ( ~ i ( m + 2 - r ) ( a + l - r ) ) E tt=l \ r = l

× ( L _ , ) ..... L ......

ifm>~0,

}+

where the coefficients ao and at are finite and the sum is over sets 1= [i~ ..... i,, I with n finite and the integers (i>2; we say that the operator is of level a and that a term all ] ( L _ , ) " - ' / I is of order Ill. We also say an operator is well-defined if it can be reduced to the form (11) by commuting all (L ~)~terms (of any power a) through to the right and if moreover, for any n, the operator can be reduced to the form

ao(L_t)~'+

+OCCL_,) . . . . . .

j) ~-Itj

')

(12)

by a finite number of these reordering operations. H e r e O ( ( L ~)'") means a sum of monomials of the form

AM,(L (9)

Y, a t L _ I ( L It+ ~,l

,)~'.'~le(L_l)"2...,~l,(L_l)"~M,.+,,

(13)

where A is a scalar, the M, are either 1 or products of

L ,withi>~2. anda~+...+a,<~m. T h i s m e a n s i n p a r 57

Volume 273, number 1,2

PHYSICS LETTERS B

ticular that if two operators are well-defined then their product is also well-defined. The naive generalisation of expressions (7) to V is not well-defined. However, a well-defined generalisation can be obtained in the following way. First, rewrite the expressions (7) by commuting all L ~ operators to the right, with no other reordering, This gives

o,,,~(t)=

y~ ~,~......~ , ( p , r - ' ) ...kr k~>~2

kl

12 December 1991

L_k~...L_~,(L

,)N-k,- .... k,-IihN, l(t) )

the expression LiOx,~lhN.~(t)). That is, Qk,._.k,(N, t, t - ~) = 0 for all integers N>~ ( [',= ~ki ) + 2,

in

and so the polynomial Q must be identically zero. Similarly, L2C~.,l(t)lh.,~(t) ), L~(~L,t,(t)lh~,t~(t) ), and L2(~,~,(l)]h~j,(t)) are all zero; hence (~Z~(t) × ]h,.~ ( t ) ) and (qj,(t) I hl,~,(t) ) are Virasoro singular vectors, It is now easy to see that a vector of the form

X~,,...X~, Ih) , O',e(t)=

Z

Px ...... x-,(q, t)

/,t ...k,

ki>~2

X L_kl ...L_,,( L_ ~)u-,_k,

(14)

where P4 ......,~(p, t) is defined for p>~E,k,, in which range it is a polynomial function in p and t. We analytically extend Pk,,...*, (P, t) to arbitrary p and define operators in Vby

(~,.~(t)=(L_~)"+

~

~

P~ ......, , ( a , t - ' )

ki >~2

× L ,~...L_k~(L 1) ~ ",

(19)

where each X o = G . ~ ( t ) or G,o(t), will be singular provided that, for each r f r o m 1 to n, ifX~ --- (%.~ (1) then h+[TZlai=h~,.~ and i f ) ( . = Cq.a,(t) then h + E',-{ ai=hl.~, To make use of these expressions, we first need to show that if X is a well-defined operator of level 0, and ifXl h ) is a Virasoro singular vector in if(h, c), then X is a scalar multiple of the identity. Suppose this is not so, and expand Xas a sum over canonically ordered partitions: X=ao+

¢~=2 kt +...+A'~ =~:

(18)

~

1~ [ir...l,i l 'j Ir~...~il ~2

alL_I(L_L )111.

(20)

for any complex numbers a and b. Now, when applied to the vacua, these operators create Virasoro singular vectors in the modules V(h,.~ (t), c(t) ) and ff(h~.~,(t), c(t) ). To see this, consider the vector

If/--{/,, ..... ix, i~} and J = {j~, ...,jx,J~} are two canonically ordered partitions, let I > J if Ill > IJ], and if III = IJI let I > J if for some k we have ik >jk and i/=j/ for all l with l < k. Then let I' = {i',, ..., i'~ } be the lowest partition with non-zero coefficient in X (that is, a r ~ 0 and if I < I ' then al=0). Then, letting I" = l~j, we have that the coefficient of L_r.(L_t)lr'l-r~+~lh) in L , _ X I h ) is non-zero, contrary to our original assumption. Now it follows from eq. (5) that

LI (~,~(t)lh,,l (t) ) ,

h~,~ + a = h ~ . j ~ a ' = - a or a' = a + 2 t - L

(~l./,(t)= (L

[)"-}-

~

E

¢t=2 kl +...+kr=tt

PLI.....kr(D, t )

ki>~2

×L_k....L_k~(L

~)~' ",

(15)

(16)

expressed as a sum & t h e form

Z

hw,+b=h~./,.~b' = - b o r b' = b + 2 t ,

Q~,.,k,(a, t, t-~)

×L_~,...L ~ ( L _ ~ ) . - ~ , - .... k~ llk..~(t)).

h~,~ =h~.~,~t(a++_ 1) = (1 ± b ) . (17)

Each Qk,..,k~(a, t, t - 1 ) is a polynomial, obtained as a sum of multiples of polynomials P~....x.;(a, t-~ ) with 2 ~= I k tI ~ ( ~ It"= 1ki ) + 2 . Thus if N is an integer larger than ( Y',_~ k, ) + 1, we have that Qx,....x~(N, t, t -~ ) is the coefficient of

Hence (~,~ (t) (% ~(t) I h ) is a singular vector at level 0. Since

('_,,, (t) e,., (t) = 1 + 0 ( ( L _ , )

-2) ) ,

(22)

and since similar observations apply to G_~,(t) × G,/,(t), we have that

(' , . , ( t ) f , , l ( t ) = l , 58

(21)

el._,,(t)G,~,(t)=l.

(23)

Volume 273, number 1,2

PHYSICS LETTERS B

12 December 1991

That is, the operators ( 15 ) are analytic extensions of the BSA operators (7). (This is not obvious from their definition.) It also follows that if h=ho.~(t) = h w , ( t ) , w i t h t(1 + a ) = 1 + b , then

Thus, for generic (h, c), the vector I h ) in if(h, c) lies at one vertex of a commutative diagram which takes the form of an infinite rectangular lattice whose points correspond to Virasoro singular vectors and whose edges correspond to operators of the form (,,~ and G,t,. (See fig. 1.) Next we consider V(h, c) when c = c ( t ) and h = h~,,q(t) for some positive integers p and q. We have

(_..~ (t) G.-~,-2(t) ¢.+2.~ (t) C w , ( t ) l h )

(25)

a=p-

( q - 1 )t-~ ~hp.q(t) =h..~ (t) ,

is a singular vector at level 0, and so we have the identity

b=q-

( p - 1)l~hp.q ( t ) = h , j , ( t )

A similar argument shows that (-p.l (t)Op.~ (t) = 1 = ¢l._q(t)O~.q(t) for positive integers p and q, and so ('p.,(t)=Or,.~(t) .

(~.,(t)=O,.q(t)

.

(24)

('l.,(,+.)+, ( t ) ¢ . . l ( t ) = e . + 2 . , ( t ) ¢ ~ . , ( , + . ) _ l ( t )

.

(27)

Hence the vectors

.

r,,+(u-~),-,.,(t)(',,+(~

(26)

3. , ~(t)...

X (:p--(q--I)t-I,l ( [ ) ] h : , . u ( t ) ) .

(28)

O-a+2l-l l

Oa,1

01,ql+a)-i

(-Ql,tl

Fig. 1.

59

Volume 273, number 1,2

PHYSICS LETTERS B

O,,,+;,.,,,+,_,(t)O,,,,(t)

and ('I,q+ (p--I),([)

=O~,.2(,,,+,)_q(t)O,,,_;,,,,+,

(:l,q+ (p--3)t ([)""

X el.q_ (p_ I >,(t)lh;,.,

(t)),

(29)

are V i r a s o r o singular vectors at level

pq in fr.(hp.u (t),

r=O

~), , , ( t )

x O;,.q(t)lh;,.,,(t)),

(30)

and

q+(p-I)z(t)".(t,-q-(;,

l),(/)

(31)

[I r=O

,>, , , ( t ) e , , + ~ _ ~ > , _ , , ( t ) . . .

I?

(;{.;+;,)+(m-q

2r); l l(/)

I

x F[ c,.,,+(,,_, :,.,,(t)

(36)

r= 0

and

1,,(,).

1 ,,(;)

(32) Note that, although the individual operators on the right-hand side of eqs. (32) do not generally belong to the Virasoro enveloping algebra, the two products do: the equations describe operator identities, not merely relations that hold to O ( ( L ~)-~). This means that (28) and (29) are in fact two equivalent expressions for the singular vector t),,q(t) in the Virasoro algebra Verma module V(hf,,q(t), c ( t ) ) . These general formulae for the singular vectors of the Virasoro algebra are our main results. An immediate application of these results is an explanation of the regularity of the Virasoro algebra's Verma module embeddings, and hence of its character formulae. For example, take a discrete series representation V(h;,.q(t), c(t) ) with t= ( m + 1 ) / m , for some integers m, p, q with m>~2 and ldq<~p<~ m1. Its singular vectors fall into an embedding pattern of type Ill in the Feigin-Fuchs classification [ 3,4 ]. The regularity of this pattern derives from two types of identities, of which the simplest examples are

O,,,+;,.,,,+l+~(t)

60

q

J~t -

X C<;,_c~_,~,-, , (t)

and

(35)

where for non-commuting operators we set ',= ~,4, =AL...A,. The second identity follows from eqs. ( 32 ) and (26). We have

are singular vectors at level O, we have that

= 03,,,+,,.~(t)O,,,+,,.,,,+

(:,,,,+,,>+<,,,+,-2,-,.,(t),

O,,,+p.m+ 1_q([) Op.q([)

X O;,.q(t)[ h;,.,;(t)) .

= e,.,+,,,_

(34)

(See fig. 2. ) The first of these follows directly from eq. (32); both sides are equal to 1]

(~_;,+(~_l),-,.~(/)...(<_p_(~

o;,.,,(t)=e,,+(~

,~(t) .

;H+q

c(t) ). But, since

(<1,

12 December 1991

,_~ (t)0;,.~

(t)

(33)

O;,.(,,,+l) ,,(t)O .... ,...... , ~(t) /)--

=

[

H r=O

(ll,[2('"+l}--ql+(/'--1--2,');([)

;;; -- q

x

17 ((' .... ;,>+~ ..... ;-2,-),-,.l(t) .

r=O

(37)

Then eq. (26) shows that the relevant quadrilateral in fig. 2 can be refined to an (' operator commutative diagram in the form o f a p X ( m + 1 - q ) rectangular lattice, with the right-hand sides of (36) and (37) each represented by two of the outer edges. The literature now contains quite a few results about Virasoro singular vectors, and so we conclude with a brief comparative discussion. Perhaps it is worth remarking that the Kac determinant formula [1,2] already provides not only an existence theorem, but also an algorithm for calculating each singular vector v;,,~(t): the singular vector is the zero eigenvector of the inner product matrix M;,q(hf,.q(t), c(t) ) at generic t, and the inner product matrix can be obtained from the Virasoro algebra's commutation relations. Bauer et al. [ 15 ] gave recursion relations which define a chain of vectors of levels 0, 1,2, .... pq in the Verma module V(hv.q(t ), c(t) ); the last of these vectors is the singular vector up.q(t). While these results have not yet led to explicit expressions

Volume 273, number 1,2

PHYSICS LETTERS B

12 December 1991

Ihp,q(m))

Op,,

Om-p,m+l-q

Op,2(m+l)-~ O:m-p,q

Om+p,rn+l-q

03rn-p,m+l -q

02m+p,q

Fig. 2.

for the singular vectors, they might well eventually do so: they might also - as Bauer et al. suggest - lead to a new proof of the Kac determinant formula, and perhaps also of the Feigin-Fuchs classification. This would certainly be valuable: an intrinsic method of proving determinant and character formulae may well be necessary in understanding the representation theory of some of the Virasoro algebra's extensions, In any case, Bauer et al. have described an interesting algebraic structure within the Verma module, which needs to be better understood; the connections with W-algebra theory and integrable models are particularly fascinating. Our expressions for the Virasoro singular vectors in the modules P(h~,.q(t), c(t) ) are simple and completely explicit. One can easily calculate the projection of one o f these vectors to the Verma module

I'(h¢,.q(t), c(t) ). However, we still do not have general explicit expressions - that is, expressions given solely in terms of elementary functions and enveloping algebra elements - for the singular vectors in the Verma modules. Such expressions would certainly be of some intellectual interest. On the other hand, we expect that the product formulae (32), together with the relations (23) and (26), will actually prove much more useful in any theoretical applications. For example, it seems unlikely that the Verma module embeddings can be understood so readily without use of the product formulae (32). (Compare the formulae of Malikov-Feigin-Fuchs [ 12] for the sl(n) singular vectors: the Verma module embeddings can easily be read off from the complex exponent formula (19), but are thoroughly obscured in eq. (20).) To summarise: the Virasoro singular vectors de61

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rive f r o m a s i m p l e algebraic s t r u c t u r e w i t h i n the m o d u l e s V(h, c), in w h i c h the analytically c o n t i n u e d BSA o p e r a t o r s are the a n a l o g u e s o f the p o w e r s o f s i m p l e roots used by M a l k o v - F e i g i n - F u c h s in the K a c - M o o d y case [ 1 2 ] . We suspect that s i m i l a r structures u n d e r l i e the highest weight r e p r e s e n t a t i o n theory o f the V i r a s o r o algebra's extensions. I a m v e r y grateful to P. G o d d a r d , H. K a u s c h , G. Watts and J.-B. Z u b e r for helpful discussions. T h i s w o r k was s u p p o r t e d by an S E R C A d v a n c e d Fellowship and by the K n o x - S h a w R e s e a r c h F e l l o w s h i p at Sidney Sussex College, C a m b r i d g e .

References [ l ] V. Kac, in: Proc. Intern. Cong. of Mathematicians (Helsinki, 1978). [2] B.L. Feigin and D.B. Fuchs, Funct. Anal. Appl. 16 (1982) 114.

62

12 December 1991

[3] B.L, Feigin and D.B. Fuchs, Funct. Anal. Appl. 17 (1983) 241. [4] B.L. Feigin and D.B. Fuchs, Reports of the Department of Mathematics (Stockholm University, 1986), in: Representations of infinite-dimensional Lie groups and Lie algebras, eds. A. Vershik and D. Zhelobenko (Gordon and Breach, London, 1989 ). [5] A. Rocha-Caridi, in: Vertex operators in mathematics and physics, eds. J, Lepowsky el al. (Springer, Berlin, 1984 ). [6] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [7] J. Cardy, Nucl. Phys. B 270 (1986) 186. [8] A. Cappelli, C. Itzykson and J.-B. Zuber, Nuel. Phys. B 280 (1987) 445; Commun. Math. Phys. 113 (1987) 1. [9] M.P. Mattis, Nucl. Phys. B 285 (1987) 671. [ 10] R.P. Langlands, Commun. Math. Phys. 124 (1989) 261. [ 11 ] A. Kent, Phys, Lett. B 269 ( 1991 ) 315. [12] F.G. Malikov, B.L. Feigin and D.B. Fuchs, Funct. Anal. Appl. 20 (1986) 103. [ 13 ] L. Benoit and Y. Saint-Aubin, Phys. Lett. B 215 ( 1988 ) 517. [ 14] M. Bauer, Ph. Di Francesco, C. Itzykson and J.-B. Zuber, Phys. Lett. B 260 (1991) 323. [ 15 ] M. Bauer, Ph. Di Francesco, C. Itzykson and J.-B. Zuber, Nucl. Phys. B 362 (1991) 515.