Weyl-Kac Character Formula Research Articles

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.

On the Wess-Zumino-Witten models on the torus

Abstract We discuss the Ward identities of the Wess-Zumino-Witten models on Riemann surfaces and point out some ambiguities in the description of the zero modes of the currents. In the case of the torus, we show how to describe them and we write the Ward identities in such a way that they become complete. We examine in detail how the Ward identities are related to the Kubo-Martin-Schwinger condition. As an illustration of this formulation, we present a new proof of the Weyl-Kac character formula. The proof essentially relies on the mixed Virasoro × Kac-Moody Ward identities and explains the relation of the heat equation on the group manifold to the Weyl-Kac character formula.

The Weyl-Kac character formula and power series identities

The Weyl-Kac character formula and power series identities

Lie algebras and classical partition identities

In this paper we interpret Macdonald's unspecialized identities as multivariable vector partition theorems and we relate the well-known Rogers-Ramanujan partition identities to the Weyl-Kac character formula for an infinite-dimensional Euclidean generalized Cartan matrix Lie algebra.