Local systems of twisted vertex operators, vertex operator superalgebras and.pdf

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2017-04-06 发布于江苏
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Local systems of twisted vertex operators, vertex operator superalgebras and.pdf

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Local systems of twisted vertex operators, vertex operator superalgebras and

a r X i v : q - a l g / 9 5 0 4 0 2 2 v 1 2 5 A p r 1 9 9 5 LOCAL SYSTEMS OF TWISTED VERTEX OPERATORS, VERTEX OPERATOR SUPERALGEBRAS AND TWISTED MODULES HAI-SHENG LI Abstract. We introduce the notion of “local system of ZT -twisted vertex operators” on a Z2-graded vector space M , generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of ZT -twisted vertex operators on M has a vertex superalgebra structure with an automorphism σ of order T with M as a σ-twisted module. Then we prove that for a vertex (operator) superalgebra V with an automorphism σ of order T , giving a σ-twisted V -module M is equivalent to giving a vertex (operator) superalgebra homomorphism from V to some local system of ZT - twisted vertex operators on M . As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite- dimensional Lie (super)algebras and we prove the complete reducibility of ZT - twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra. 1. Introduction In [Li], motivated by [G] we introduced the notion of local system of vertex operators and proved that any local system of vertex operators on a super vector space M has a vertex superalgebra structure with M as a module, so that if V is a vertex (operator) superalgebra, then giving a V -module M is equivalent to giving a vertex superalgebra homomorphism from V to some local system of vertex operators onM . Similar to the notion of local system of vertex operators, the notion of commutative quantum operator algebra was introduced in [LZ1-2]. For a vertex (operator) superalgebra V , in addition to having module theory, more generally we have twisted module theory ([D], [FFR], [FLM]). In this paper, generalizing the notion of local system of vertex operators ([G], [Li]), we introduce the notion of what we call “local system of ZT -twisted vertex operators” on a Z2-graded vector