Modular invariant representations of infinite-dimensional Lie algebras and superalgebras

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(lambda) of a Kac-Moody algebra unk 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 unk, this class includes modular invariant representations of arbitrary rational level m = t/u, where t unk Z and u unk N are relatively prime and m + g >/= 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 unk = A(1) ((1)), 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 A(1) ((1)) 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(0, 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.