Modular Invariant Representations Of Infinite-dimensional Lie Algebras And Superalgebras

Modular Invariant Representations of Infinite-Dimensional Lie Algebras and Superalgebras Victor G. Kac, and Minoru Wakimoto PNAS 1988;85;4956-4960 doi:10.1073/pnas.85.14.4956 This information is current as of March 2007. This article has been cited by other articles: www.pnas.org#otherarticles E-mail Alerts

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Proc. Nati. Acad. Sci. USA Vol. 85, pp. 4956-4960, July 1988 Mathematics

Modular invariant representations of infinite-dimensional Lie algebras and superalgebras (Kac-Moody algebras and superalgebras/Virasoro algebra/modular functions and theta functions/asymptotic dimension)

VICTOR G. KACt AND MINORU WAKIMOTOt4 tMassachusetts Institute of Technology, Cambridge, MA 02139; and tMie University, Tsu 514, Japan Communicated by Shlomo Sternberg, March 16, 1988

In this paper, we launch a program to deABSTRACT scribe and classify modular invariant representations of infinite-dimensional Lie algebras and superalgebras. We prove a character formula for a large class of highest weight representations L(A) of a Kac-Moody algebra g with a symmetrizable Cartan matrix, generalizing the Weyl-Kac character formula [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70]. In the case of an affine g, this class includes modular invariant representations of arbitrary rational level m = t/u, where t E Z and u E N are relatively prime and m + g 2 g/u (g is the dual Coxeter number). We write the characters of these representations in terms of theta functions and calculate their asymptotics, generalizing the results of Kac and Peterson [Kac, V. G. & Peterson, D. H. (1984) Adv. Math. 53, 125-264] and of Kac and Wakimoto [Kac, V. G. & Wakimoto, M. (1988) Adv. Math. 70, 156-234] for the u = 1 (integrable) case. We work out in detail the case g = A"), in particular classifying all its modular invariant representations. Furthermore, we show that the modular invariant representations of the Virasoro algebra Vir are precisely the "minimal series" of Belavin et al. [Belavin, A. A., Polyakov, A. M. & Zamolodchikov, A. B. (1984) Nucl. Phys. B 241, 333-380] using the character formulas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1984) Lect. Notes Math. 1060, 230-245]. We show that tensoring the basic representation and modular invariant representations of AV') produces all modular invariant representations of Vir generalizing the results of Goddard et al. [Goddard P., Kent, A. & Olive, D. (1986) Commun. Math. Phys. 103, 105-119] and of Kac and Wakimoto [Kac, V. G. & Wakimoto, M. (1986) Lect. Notes Phys. 261, 345-371] in the unitary case. We study the general branching functions as well. All these results are generalized to the Kac-Moody superalgebras introduced by Kac [Kac, V. G. (1978) Adv. Math. 30, 85-136] and to N = 1 super Virasoro algebras. We work out in detail the case of the superalgebra B(O, 1)(1), showing, in particular, that restricting to its even part produces again all modular invariant representations of Vir. These results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations. Section 1. Let g be a complex (in general infinite-dimensional) Lie algebra or superalgebra, and let E be a fixed element of g (the energy operator). We call a g-module V modular invariant if the following two properties hold: (i) E is diagonalizable: V = EAEspeCE VA, with finite-dimensional eigenspaces VA, and (ii) trve-2wrT(E+a) : -A(dim VAje-2XA+a) (where T E C, Re T > 0) is a holomorphic modular function in iT (of weight 0) on the upper half-plane with respect to some L(N) C SL2(Z), for some a E C. The number a is called the modular anomaly of V. Note that a + SpecE lies in N-1Z and is bounded below, so that V is a positive energy module. Recall that V is called a positive

energy module if (i) holds and SpecE C h + N-1Z+, h E SpecE, for some h E C and N E N; the number h is called the trace anomaly of V. (Here and further, Z+ = {0 1, 2, .. .}, N = {1, 29 . . .}.)

CONJECTURE 1. Let g be a Lie algebra or superalgebra of finite growth and let E be an ad-diagonalizable element of g with finite-dimensional eigenspaces, such that g has only obvious (adE)-invariant ideals. Let V be an irreducible positive energy g-module. Then,

trve-2xfTE

-

ATB/2e C/12T as T,0O

[1]

for some constants A, B, and C. If E is rational and B = 0, then V is modular invariant. This conjecture is true for the Virasoro, Neveu-Schwarz, and Ramond algebras and also for the rank 2 affine algebras (see below). We call the triple of numbers (A, B, C) the asymptotic dimension of V and write: asdim V. Note that for a modular invariant module, expression 1 holds with B = 0, C > 0. Note also that the knowledge of asymptotic dimensions allows one to compute branching coefficients (see ref. 1). Section 2. Here and further we use notations and basic definitions of ref. 2, unless otherwise stated. Let g = g(A) be a Kac-Moody algebra with a symmetrizable generalized Cartan matrix A and let b be the Cartan subalgebra of g. Let R (resp R+) C b be the set of real (resp positive real) coroots of g. Let ra denote the reflection in b* with respect to a and W = (rtIa E R) be the Weyl group. We denote by w(A) the usual action of w E W and by w.A = w(A + p) - p the "shifted" action, where p is the sum of fundamental weights Ai. Let K = {A EE b*1(A, a) 2 0 for all but a finite number of a E R+ and (A, a) # 0 for an isotropic coroot a} (see ref. 3). (K differs slightly from the complexified Tits cone of ref. 2.) Here and further, for a E C, we write a > 0 if either Re a 2 0 or Re a = OandIma 20. Fix A E )*. Let RA = {a E RI(A, a) E Z}, RA = RA n R+, IA = {a E R+4'a not equal to a sum of several roots from R+}, WA = \Vala E RA). Let L(A) (resp M(A)) denote the irreducible highest weight g-module (resp Verma module) with highest weight A. Let 09 denote the subcategory of the category e consisting of g-modules all irreducible subquotients of which are L(A) with A E -p + K. Let A E -p + K, and let CA be the subcategory of 05 consisting of g-modules all irreducible subquotients of which are of the form L(w.A) with w E WA. Then 0A depends only on the orbit WA.A, and any g-module from C9 decomposes uniquely as a direct sum of g-modules from CA, A E -p + K (ref. 3). Thus, for a g-module V from 0C, we may= define prAV E CA as the CA-summand of V. Let P+ {A E b*I(A, a) E Z+ for a E R+}. Given A, IL E

-p + K such that the set W(,u - A) n P+ is nonempty and hence contains a unique element A, define the translation functor TA: CA -- C0. by (see refs. 3-5): TA,(V) = pr,(L(A) 0 V), V E CA. This is an exact functor whose main property is given by Lemma 1. LEMMA 1. Let A, u E8 -p + K be such that W(x - A) n P+

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4956

Proc. Natl. AcadJ Sci USA 85 (1988)

Mathematics: Kac and Wakimoto = {A}. Suppose thatfor any a E RA, (A + p, a) # 0 and that (A + p, a) > 0 (resp < 0) implies that (,u + p, a) 2 0 (resp 0). Then TA' (M(w.A)) = M(w.pu) for w E WA. Proof: We may assume that (A + p, a) > 0 for all a E R4, so that (Ii + p, a) 2 0 for all a E R+. Then, as usual (see refs. 3-5) the only thing to show is

part is removed; e.g., W = (rj, ., r1), = e 1 Ca, A C * is the set of roots of A, etc.; for A E I*, we denote by A the restriction of A to A. We consider here the following special class of admissible values of A. Fix y E W and k = (ko, ..., ke) with ki E ai(ay -1Z, such that /3i := ki + y(ay) E R+, and let

A + v = w(,u) for w E WAand a weight v of L(A) E> w(,u) = tt. We identify b and I* via the standard invariant bilinear form (.1.). We have (v| v) = (AlA) + (,ul40) - 2(Akw(A)). But w(,u) = ,u -IaERt naa, where na 2 0 (see, e.g., section 3.14 of ref. 2), hence, (Alw(pt)) c (A|,u) with equality iff w(pu) = ,u. Hence v) 2 (#M - AlIu - A) with equality iff w(,u) = pi. Since (vl v) (vl c (Ii - Alli - A) = (AlA) (section 11.4 of ref. 2), Lemma I is E proved (cf. ref. 5). THEOREM 1. Let A E -p + K be such that (A + p, a) > 0 for all a E R'. Then

ch L(A) = > E(w)ch M(w.A). WEWA

p,;k {130, =

ch L(A) = E m(A, w)ch M(w.A), WEWA

where m(A, w) E Z, m(A, 1) = 1. [3]

el;

U =

1 + Eayki. i=O

m = t/u for some t E Z relatively prime to u and m + g (b) Let m satisfy Eq. 4. Then

pyi {Y.I (n,

ki(m

=

+

u-1g. [4]

g))Ai -,

[5]

a where ni E 4, Jj=O aj'ni = u(m + g) - g. For A as in Eq. 5, we write AO = Y;,_o nAj. Remark 3: If A 8E Pm with m E Z, then m E 4+ and A E P+. However, for other 1A there are admissible A § P+ even of level 0. There are no other 11A for g of type A(') In order to state the character formula, introduce some notation. For A E f)* of level m + -g put

For a E H1A there exists ,u E -p + K such that W(.u - A) n P+ # 0, (A + p, a) = 0, but (it + p, /B) > 0 for all /3 E -

C(m)

HIA\{a}. Applying Tr to the exact sequence M(A)/M(ra,.A)

we obtain T"(L(A)) = 0 by Lemma 1. Hence, applying TA to both sides of Eq. 3, we obtain, by Lemma 1, o = EWCEWA m(A, w)ch M(w.A). Since the coefficient of ch M(w.,u) in this equation is m(A, w) + m(A, wra) and since WA [ = (raja E HA), we deduce that m(A, w) = E(w). Remark 1: At least implicitly, Theorem I for the finitedimensional g is contained in ref. 4. For A E P+, Theorem I turns into the Weyl-Kac character formula for integrable L(A) (see ref. 7). Remark 2: Given A E - p + K, ch M(-p)f ch L(A) is WA_ invariant iff (A + p, a) > 0 for all a E R'. This condition does not imply in general, however, that ch L(A) is Wkinvariant. For each a E IA there exists a singular vector Va E M(A) of weight ra.A, the highest weight vector of M(ra.A) = U(g)va C M(A) (see ref. 6). The following result, known for integrable L(A) (see refs. 2 and 7), follows immediately from the proof of Theorem 1. COROLLARY 1. Let A satisfy conditions of Theorem 1. Then L(A) = M(A)/I;aCHnA U(g)va. From refs. 3 and 6, we deduce the following. PROPOSITION 1. Let g be a Kac-Moody subalgebra of g whose triangular decomposition is consistent with that of g. Suppose that A satisfies conditions of Theorem 1 and that for any irreducible subquotient L(fr) of the g-module L(A) viewed as a 4-module jL satisfies conditions of Theorem 1 for g. (Here andfurther, overdots signify objects related to [1 g.) Then the 4-module L(A) is completely reducible. Section 3. Here we apply Theorem I to a Kac-Moody algebra g associated to an (f + 1) x (e + 1) affine matrix of type X(k). In this case, Int K = {A E tb*jRe (A, c) > 0}, where c = J!=O Aala, is the canonical central element of g. A E Ib* is called admissible if it satisfies the conditions of Theorem I and the dimension of the C-span of HA is e + 1. Note that the level (A, c) of an admissible A is a rational number that is greater than -g, where g = (p, c) is the dual Coxeter number. Further on, an overbar signifies that the 0th

. *

Then u E N. Let Pk.denote the set of all admissible A of level m with 11A = fly;k. and let Pm = UYj Pyk. PROPOSITION 2. (a) Pykj # 0 iff

[2]

Proof: We have (see refs. 3 and 4)

4957

=

md

m + g'

L(A) -* 0 (6),

hA =

(A + 2p A) + bA, 2k(m + g)

SA = k(hA - 2 c ),

[6]

where d = dim g(XN) and bA =m/(m + g)) ((d/24) - iPl2/ 2gk) (=0 if k = 1). Given A 8E * and m E8 N, introduce the theta function OA,m (T, z) by a series that converges to a holomorphic function for Re T > 0 and z E f:

01,m(T, z) =

E -r' exp[ 2 TmT(

QYP)

+

(

)]

THEOREM 2. Let A E PYk. Define /3E I* by (P, y(ai')) =ki, i= 1, . . ., e. Then for Re T > 0 and z E- such that (z, a) E Z 4> (/3 + v(z)la) E uZ, a E A, we have the following. (a) XA(T, z) = trL(A)e2,T(d-z-sA)

>E e(Wy)Ouw(°o+p)-u(m+g)pu2(m+g)(T, u'z)

weww wE W

In particular, for an energy operator E = -d + z, where z is rational, i.e., z E 1 l Qda, the g-module L(A) is modular invariant.

(b) asdim L(A) = (a(A) n

sin

-(3

+

v(z)ja)/siniiT(z, a),,g(m)0,

where a(A)

=

17 2 sin 1T(AO + pla)/u(m + g), u-IP/MI-1/2(m + g)-E/2e(y) aEA+

4958

Mathematics: Kac and Wakimoto

Proc. NatL Acad ScL USA 85

and

0n,m(T, Z) g(m)

=

4)(m + (1 - u-2)g)

(dim

m+g

=

I

,ffpm

-Ob,a(T,

u

1z)

,-

(21a) 1/2AA EFpRm

e(-iirb+b'/a) - e(-ib+b+/a) 2ie XA(T, z); 2i

asdim L(A) =

g)(3811)].

Proof: The proof of Theorem 2 b and c uses transformation properties of theta functions. O Remark 4: If A E P+, then asdim L(A) is independent of z (cf. ref. 8). CONJECTURE 2. For E = -d + z with rational z, the gmodule L(A) is modular invariant iff A is admissible. (This conjecture holds for the rank 2 affine algebras. It is also easy to show, using ref. 6, that if L(A) is a modular invariant module, then m + g is a positive rational number.) The following tensor product decomposition formula is useful for applications. PROPOSITION 3. Let g be a simply laced affine algebra and let A E Pji. Let A E P+ be of level 1. Then

k;

z)

sin irz, 0, 3 - 6/a)

One can show that an A(')-module L(A) is modular invariant iff one of the following equivalent properties holds: (i) A E Pm with admissible m; (ii) asdim L(A) = (A, 0, C); or (iii) asdim (L(A) = (A, B, C) with C < 3. Furthermore (i = 0, 1),

I

XA(T, Z)XA(T, Z) =

Osn':t+3u-2

x(m+2), n+ln +1(T)Xk.+l*;n'(T, Z), [7]

n'-n+imod2

where

and p, q EE N are relatively prime and such that m + 2 =

q/(p - q). Putting X(O) = Xp, X(1a) = X2A0 + X2AI, we have for e = 0 or

where

1/2,

=

X(E)(T, Z)XA(T,

77(iT)-t E

weW

VEMe (w)e-1rT(m+9)mg~)u

m+g+l

Z) =

M+gl

Remark 5: Proposition 3 holds for twisted g, with rq(iT)e replaced by G(iT) of page 214 of ref. 8. Remark 6: Proposition 3 holds for g - B with XA replaced by 'q(iT)-t'q(iT)X(6), e = 0 or 1/2, where X(o) : = XAe; X(12) : = XAO + XA1, and 77o(T) := 7(T)2/17(2T); r112(T)

B(V/2T)rn(2T)/7j(T).

Example 1: Let g be of type A(') Then the level m of an admissible A is of the form m = t/u, where u E N, t E Z is relatively prime to u and 2u + t - 2 2 0; such m is called admissible. The set Pn of all admissible A (considered mod C8) having an admissible level m is =

+2)

(,ms+-)(T)= i(iT)r'(0prqspq(T, 0) - epr+qsspq(T, 0)),

Y;k

r

((2/a)11(sin iX u

bA, A(T)X,

A+A-,u(=Q

bA, A(T)

{Am;k;n := (m - n + k(m + 2))Ao + (n - k(m + 2))A11 n, k E Z+, n c 2u + t - 2, k s u - 1}.

z

Osn':t+4u-2 n'-n+1-2emod2

Xn

+1,n)6+,E(T)XAm+2.n ,(T, z),

-rvs;e =,6(iT)-l(O(pr-qs)2,pql2(T, 0) - O(pr+qs)/2,pq/2(T, 0)),

and p, q E N are such that p - q is even, (p - q)/2 and q are relatively prime, and m + 2 = 2q/(p - q). Section 4. Recall that the Virasoro algebra Vir is a Lie algebra with basis dj(j E Z), c and commutation relations [dj, dk] = (j - k)d1+k + [(j3 - j)/12]6j,-kc, [dj, c] = 0. We take for the energy operator E = do and denote by V(c, h) the irreducible positive energy Vir-module with trace anomaly h and conformal anomaly c (= the eigenvalue of c). Given n E Q\{0, -1}, we can represent it uniquely in the form n = q/(p - q), where p E Z, q 8E N, and p and q are relatively prime, and let

SA =

b+ := u(±(n + 1) - k(m + 2)),

(b21/4a)

n;a

- 8

Then the character XA(T, z) := trL(A)e2 T ~d(I2)Za1SA] is expressed in terms of theta functions

1

6 ,1

gn n(6+ 1) pq a2 - (p q)2 for a 8 C. =

u2(m + 2),

[8]

where

Fix such A = Am;k;n and let a :=

-z)

iTz) =

,u)X,,(T, z),

P',t

I

Ob,a(T, u

01,2(T, z) - 1e,2(T, z)

XA(T, Z)

XA

a(A,

where for E we define 13' by (13'Iy'(ai)) = k, i = 1, (, and let a(A, u) = e(yy') lwEC E(w)exp(2iri[-(m + g)-l(w(AO + p)I(,u0 + p)) + (A° + plff) + (AO + p113) - (m +

fePm+l

E C,

we have the following transformation law:

iI'+IlM*/u2(m + g)MI-"J2

XAXA =

z

as follows:

x

A

e-21rmT(j2+jz)

jEZ+n/2m

b

where b = g-lkg(= 1 if k = 1). (c) Provided that g is simply laced, we have the following transformation law:

XA(T-1, -iTz)

I

:=

(1988)

4pq

Using the Feigin-Fuchs character formulas (see ref. 9) and their transformation properties (see ref. 10), we obtain the following.

Mathematics: Kac and Wakimoto

Proc. Natl Acad Sci USA 85 (1988)

PROPOSITION 4. (a) The asymptotic dimensions of the Virmodules V(c, h) are as follows. Case I. c = cn, h = hn;a, with n ECl 0, n > 0: (i) asdim V(c, h) = (2irsr, 3, 1) if a E pZ+ U qZ+, where r, s E N are such that a = spq - r, r

-

4959

(c) For V(c, h) from (b) one has (cf. Example 1):

trv(,h,~p)e-T(4+cn/24)

=

XP (T)o

Using asymptotics gives the following. COROLLARY 2. (a) There are exactly five positive energy irreducible Vir-modules of asymptotic dimension (A, B, C) with C 5 1/2:

pq;

(ii) let a E Z+\(pZ+ U qZ+), then there exist unique r, s, k E Z+ such that a = qs - pr, 0 < r < q, kp < s < (k + 1)p,

(b) A tensor product of two irreducible nontrivial positive energy Vir-modules has afinite Jordan-Holder series iff one of them is of type i and the other is of type i, ii, or

and we have

asdim V(c, h) = (2irkr((k + 1)p - s), 3, 1) if k 2 1 asdim V(c, h) = (AN;roso 0, gn) if k = 0,

(iii) V[(-68/7), h] with h = 0, -2/7 or -3/7 (then C = 4/7).

where ro, so E N are defined by ro < q, so < p, rop - soq = 1, and A (,n);rO

(i) V[(-22/5), h] with h = 0 or -1/5 (then C = 2/5), (ii) V[(1/2), h] with h = 0, 1/16 or 1/2 (then C = 1/2).

(8/pq)"/2( )(r+s)(ro+so)sin

rro

n

sin

Remark 7:

sso

n+l

Case I1. c = Cn, h = hn;a with n E 0C, (-1/2) s n < 0:

(i) asdim V(c, h) = (2lrkr, 3, 1) if a E -pZ+ U qZ+, where r EE N, k E Z+ are such that a = -kpq + r, r

<

-pq;

(ii) let a E Z+\(-pZ+ U qZ+), then there exist unique r, s, k E Z such that a = qs - pr, 0 < r < q, -kp < s < -(k + l)p, and we have asdim V(c, h) = (1, 1, 1) if k < 0 asdim V(c, h) = (27r(k + 1)r(pk + s), 3, 1) if k

-

-

n

n(n + 2)/'

h) - a2-_(p MEa 8p q)2 8pq na

(2ir(Ial + 1), 3, 1) if a E8 Z.

asdim V(c, h) = (2irr(r - s), 3, 1) if a = r + ns, r E N, s E Z, r - s > 0, asdim V(c, h) = (1, 1, 1) otherwise.

(b) The following four conditions on the Vir-module V(c, h) are equivalent.

0Q\{0, -1}, h

- 3 1

n 2\

hn;rp-sq, 8 where r, s N, r < q - 1, s c p

=32

1

8 pq

'

+ B (- 2 E). 16

We state here only results related to the superanalogue of the minimal series. PROPOSITION 4S. (a) The following four conditions on

V6(c, h) are equivalent:

(i) C - (E) with n 8 Q\{0, -2}, h = hn;rp-qs, where r, s E8 N, r s q - 1, s s p - 1, and r - s (1 - 2e)mod 2; (ii) V6(c, h) is a modular invariant Vir.-module; (iii) asdim V.(c, h) = (A, 0, C); (iv) asdim VE(c, h) = (A, B, C) with C < 3/2.

(b) For VE(c, h) from (a) one has (cf. Example 1)

=

-

1,

i.e., V(c, h) is in the minimal Belavin-Polyakov-Zamolodchikov series (ref. 11); (ii) V(c, h) is a modular invariant Vir-module; (iii) asdim V(c, h) = (A, 0, C); (iv) asdim V(c, h) = (A, B, C) with C < 1.

gn(E)

0.

Case IV. asdim V(c, h) = (1, 1, 1) in Cases I, II, and III ifa E C\Z. Case V. c=cn, n E C\0, h = hna, a E C:

(i) C = Cn with n

(1 - q')-, Hl tTV[(-22/5),O(resp-l5)]qA-h = n+O±1(resp±|2)mod5 neN

which are the left-hand sides of the Rogers-Ramanujan identities. More generally, taking c = 1 - 3(m - 2)2/m, m = 5, 7, 9, . .., we get the left-hand sides of Gordon's generalizations of these identities (see ref. 16). It is an interesting problem to construct these representations explicitly. The Lie algebra Vir is the even part of the Neveu-Schwarz and Ramond superalgebras, denoted by Vir8, where e = 1/2 and 0, respectively (see, e.g., ref. 13 for commutation relations). We take the same energy operator do and denote an irreducible positive energy Vir8-module by V6(c, h). Given n E 0\{0, -2}, we can represent it uniquely in the form n = 2q/(p - q), where p E Z, q E N, p - q 8E 2Z, and (1/2)(p - q) and q are relatively prime, and let

(£e)

Case Ill. c = 1, h = (1/4)a2:

asdim V(c, h) =

The corresponding partition functions (see ref. 12) are of O type (A2, D6), (A4, E6), and (A6, E8), respectively.

try8 (ckehtep qde

asdim VE (cn

,

22TT(do+c(n)/24) =

hn--p-qs) - (A(

s(n); 0, g ),

where ro, so E8 N are defined by rO < q, so < p, rO - so 0 mod 2, prO - qsO = 2, and

A(E) rls;r0,s0

-

2E(8/pq)/2 sin -TIT0 n

+T

sin n + 2' 2.

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Mathematics: Kac and Wakimoto

Section 5. Let g be defined as in Section 3 and let g' denote its derived algebra. Consider the semidirect product 4 of g' with Vir such that g g' + Cdo and d = -kaodo. Then a gmodule V from the category C, on which c acts as a scalar m # -g, extends canonically to a 0-module in which c acts as a scalar c(m) and do(v) = (hA - (kao)-'A(d))v for a primitive vector v of weight A (see Eq. 6 and ref. 1). We obtain, using Proposition 1, Theorem 2b, Corollary 2, and refs. 1 and 9 the following. PROPOSITION 5. Let g be an affine subalgebra of g whose triangular decomposition is consistent with that of g and which is invariant under the action of Vir. For a g-module L(A) oflevel m (resp viewed as a 4-module, of level mi) denote by L6 (resp LO) the operators on L(A) corresponding to dn, and let Ln = Lo - Lo (we assume that m 7 -g and mi (a) (Refs. I and 14.) The 4-module L(A) extends to a module over the direct sum g' E Vir by putting dn '-*Ln, c '- c(m) , with trace anomaly for Vir equal hA - hA. (b) Assume that A is admissible and such that ,u is admissible fqr g if L(,u) is an irreducible subquotient of L(A). Given ,u E b*, consider the Vir-module U(A, ,u) = {v E L(A)jfi+(v) = 0, h(v) = j4(h)v for h 8 Then (i) with respect to Vir E g' we have the following decomposition: =

6}.

L(A)= E U(A,

f)0 0 L(A);

( c( (ii) the functions b,(T) = trU(A,,)e-21rT(do+(c are holomorphic modular functions in iT on the upper halfplane; (iii) the Vir-module U(A, it) has a finite Jordan-Holder series iff g(m) - 9("') < 1, in which case all nonzero b4 are finite sums of the Xrn) (cf. Example 1). ° The following corollary is a generalization of refs. 17 and 18. COROLLARY 3. (a) Eq. 7 is the character formula of the direct sum decomposition of the g'-module L(Aj) 0 L(A) with respect to the direct sum of Vir and of g' of type Ai). (b) For each k E Z+, k < u (the denominator of m), the Vir-module (L(Ao) 0& ZACPIk L(A))n+ decomposes into a direct sum of all modular invariant Vir-modules with conformal anomaly Cm+2, each of them appearing once. (c) Eq. 8 is the character formula of the direct sum decomposition of the g'-module L(p) 0 L(A) (resp (L(2Ao) + L(2A,)) 0 L(A)) for e = 0 (resp e = 1/2) with respect to the direct sum of g' of type A(') and Vire, and a result similar to (b) holds. Section 6. Let g be a Kac-Moody superalgebra, introduced and studied in ref. 15, and let b be its Cartan subalgebra. Let Rodd (resp RCVn) C f be the set of all odd (resp even) real coroots of g, so that (1/2)Rodd C Reven. We let W = (rcla GE ReVen) be the Weyl group and define the domain K in the same way as in Section 2. Given A E b*, we let Rodd = {a 8 RoddI(A + p, a) E8 Zodd}, Reven = {a Reven\(1/2)RoddI (A + p, a) E Z} U (1/2)Rodd, RA = Rodd U Reven, WA = (rjIa E RA). THEOREM 1S. Let A E - p + K be such that (A + p, a) > 0 for all a 8 R+. Then Eq. 2 holds for the g-module L(A). Example 2: Let g be of type B(0, 1)(1); i.e., g' = osp211(C[t, tF']) + Ce with commutation relations [a(t), b(t)]+ = a(t)b(t) ± b(t)a(t) + (Resostrb(t)da(t))c, [g', c] = 0.

Proc. NatL Acad ScL USA 85

(1988)

Then the level m of an admissible A is of the form m = t/u, where u E N, t E Z is relatively prime to u, and for t odd (resp even), 3u + t - 1 2 0 (resp 3u + t - 3 - 0). The set Pm of all admissible A (mod CS) of level m is Pm = {Am;k;n : = 2(n - k(m + 3))Ao + ((1/2)m - n + k(m + 3))A1Ik, n E (1/2)Z+, k + n E Z+, k s u - (1/2). n s u(m + 3) - 1 if t is odd; and k, n E8 Z+, k s (1/2)(u - 1), n s (1/2)(u(m + 3) - 3) or (1/2)(u + 1) sk s u - 1, (1/2)(u(m + 3) + 1) s ns u(m + 3) - lift is even}. The even part of the superalgebra g' is the algebra g' = se2(C[t, t- ]) + Uc of type A('); its simple coroots are & = (1/2)a', = a1, di where a' and ad are the odd and even simple coroots of g'. As in the Lie algebra case, an admissible g'-module L(A) of level m = t/u, viewed as a g'-module, extends to a module over the direct sum Vir 3 g', and the decomposition formula is in all cases except for those in parentheses (resp if t is even and (1/2)(u + 1) c k c u - 1): L(Am;k;n) = ;

q/(pq) hq/(p-q);(n' +l)p-(2n+ l)q(resp-(2n+ l)q+pq)

n''q-2

09 L(A(JJ2)m;2gzresp 2k-u);n'), wherep = 2t + 6u, q = t + 4u (respp = t + 3u, q = (1/2)t + 2u) if t is odd (resp t is even), and Am;k;n = (n - k4m + 2))Ao + (m - n + k4m + 2))Ai. This paper is dedicated to I. M. Gelfand on his 75th birthday. We are grateful to D. Vogan for valuable advice. This research has been partially supported by National Science Foundation Grant DMS 8508953 and by the Suurikagaku-Shinkokai Foundation. 1. Kac, V. G. & Wakimoto, M. (1988) Adv. Math. 70, 156-234. 2. Kac, V. G. (1985) Infinite Dimensional Lie Algebras (Cambridge Univ. Press, Cambridge, U.K.), 2nd Ed. 3. Deodhar, V. V., Gabber, 0. & Kac, V. G. (1982) Adv. Math. 45, 92-116. 4. Jantzen, J. C. (1979) Modulin mit einem hochsten Gewicht, Lecture Notes in Mathematics (Springer, New York), Vol. 750. 5. Vogan, D. A. (1981) Representations of Real Reductive Groups, Progress in Mathematics (Birkhauser, Boston), Vol. 15. 6. Kac, V. G. & Kazhdan, D. A. (1979) Adv. Math. 34, 97-108. 7. Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70. 8. Kac, V. G. & Peterson, D. H. (1984) Adv. Math. 53, 125-264. 9. Feigin, B. L. & Fuchs, D. B. (1984) Lect. Notes Math. 1060, 230-245. 10. Itzykson, C. & Zuber, J. B. (1986) Nucl. Phys. B 275, 580-595. 11. Belavin, A. A., Polyakov, A. M. & Zamolodchikov, A. B. (1984) Nucl. Phys. B 241, 333-380. 12. Capelli, A., Itzykson, C. & Zuber, J. B. (1987) Nucl. Phys. B 280, 445-460. 13. Kac, V. G. & Todorov, I. (1985) Commun. Math. Phys. 102, 337-347. 14. Goddard, P., Kent, A. & Olive, D. (1985) Phys. Lett. B 152, 88-92. 15. Kac, V. G. (1978) Adv. Math. 30, 85-136. 16. Andrews, G. (1976) The Theory of Partitions (Addison-Wes-

ley, London). 17. Goddard, P., Kent, A. & Olive, D. (1986) Commun. Math. Phys. 103, 105-119. 18. Kac, V. G. & Wakimoto, M. (1986) Lect. Notes Phys. 261, 345-371.