Twisted modules for vertex operator algebras and Bernoulli polynomials

Abstract

This work is a continuation of a series of papers of two of the present authors [L3], [L4], [M1]–[M3], stimulated by work of Bloch [Bl]. In those papers we used the general theory of vertex operator algebras to study central extensions of classical Lie algebras and superalgebras of differential operators on the circle in connection with values of ζ–functions at the negative integers. In the present paper, using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call “modified weak associativity.” This paper is an announcement. The detailed proofs will appear elsewhere. More specifically, the present goal is to obtain a new general Jacobi identity for twisted operators, and for related iterates of such operators, extending the previous analogous results in the untwisted setting in our papers mentioned above. As a consequence we obtain twisted constructions of certain central extensions of Lie algebras of differential operators on the circle, combining and extending methods from [L3], [L4], [M1]–[M3], [FLM1], [FLM2] and [DL]. In those earlier papers we used vertex operator techniques to analyze untwisted actions of the Lie algebra D, studied in [Bl], on a module for a Heisenberg Lie algebra of a certain standard type, based on a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form. Now consider an arbitrary isometry ν of period say p, that is, with ν = 1. Here we announce that the corresponding ν–twisted modules carry an action of the Lie algebra D in terms of twisted vertex operators, parametrized by certain quadratic vectors in the untwisted module. In particular, we extend a result from [FLM1], [FLM2], [DL] where actions of the Virasoro algebra were constructed using twisted vertex operators. In addition we explicitly compute certain “correction” terms for the generators of the “Cartan subalgebra” of D that naturally appear in any twisted construction. These correction terms are expressed in terms of special values of certain Bernoulli polynomials. They can in principle be generated, in the theory of vertex operator algebras, by the formal operator ex [FLM1], B.D. gratefully acknowledges partial support from an NSERC Postgraduate Scholarship. J.L. and A.M. gratefully acknowledge partial support from NSF grant DMS-0070800.