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Generalized Twisted Modules Associated to General Automorphisms of a Vertex Operator Algebra

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Abstract

We introduce a notion of a strongly \({\mathbb{C}^{\times}}\)-graded, or equivalently, \({\mathbb{C}/\mathbb{Z}}\)-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of a strongly \({\mathbb{C}}\)-graded generalized g-twisted V-module if V admits an additional \({\mathbb{C}}\)-grading compatible with g. Let \({V=\coprod_{n\in \mathbb{Z}}V_{(n)}}\) be a vertex operator algebra such that \({V_{(0)}=\mathbb{C}\mathbf{1}}\) and V (n) = 0 for n < 0 and let u be an element of V of weight 1 such that L(1)u = 0. Then the exponential of \({2\pi \sqrt{-1}\; {\rm Res}_{x} Y(u, x)}\) is an automorphism g u of V. In this case, a strongly \({\mathbb{C}}\)-graded generalized g u -twisted V-module is constructed from a strongly \({\mathbb{C}}\)-graded generalized V-module with a compatible action of g u by modifying the vertex operator map for the generalized V-module using the exponential of the negative-power part of the vertex operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening operators on certain vertex operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group \({\mathbb{C}/\mathbb{Z}}\) or \({\mathbb{C}}\) and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted vertex operators in general involve the logarithm of the formal variable.

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  1. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ, 08854-8019, USA

    Yi-Zhi Huang

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Communicated by Y. Kawahigashi

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Huang, YZ. Generalized Twisted Modules Associated to General Automorphisms of a Vertex Operator Algebra. Commun. Math. Phys. 298, 265–292 (2010). https://doi.org/10.1007/s00220-010-0999-6

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