In this paper, we launch a program to describe 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][69][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 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

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We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all superconformal algebras.

The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an "irrefutable proof" of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.

We extend the homological method of quantization of generalized Drinfeld--Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.Comment: 33 pages, LaTe