A modular invariance on the theta functions defined on vertex operator algebras

Abstract

1. Introduction. Throughout this paper, V denotes a vertex operator algebra, or VOA, (⊕ ∞ n=0 V n , Y, 1, ω) with central charge c and Y (v,z) = v(n)z −n−1 denotes a vertex operator of v. (Abusing the notation, we also use it for vertex operators of v for V -modules.) o(v) denotes the grade-keeping operator of v, which is given by v(m − 1) for v ∈ V m and defined by extending it for all elements of V linearly. InWe call V a rational vertex operator algebra in the case when each V -module is a direct sum of simple modules. Define C 2 (V ) to be the subspace of V spanned by elements u(−2)v for u, v ∈ V . We say that V satisfies condition C 2 if C 2 (V ) has finite codimension in V . For a V -module M with grading M = ⊕M m , we define the formal character asIn this paper, we consider these functions less formally by taking q to be the usual local parameter q = q τ = e 2πιτ at infinity in the upper half-planeAlthough it is often said that a VOA is a conformal field theory with mathematically rigorous axioms, the axioms of VOA do not assume the modular invariance. However, Zhu [Z] showed the modular (SL 2 (Z)) invariance of the space q |a 1 | 1 · · · q |a n | n tr W Y (a 1 , q 1 ) · · · Y (a n , q n )q L(0)−c/24 : W irreducible V -modules (2) for a rational VOA V with central charge c and a i ∈ V |a i | under condition C 2 , which are satisfied by many known examples, where q j = q z j = e 2πιz j and |a i | denotes the

Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness

“…This is a fundamental result for modular invariance properties of vertex operator algebras and is extended by several authors, e.g. [DLiM3], [Miy1]- [Miy3] and [Y].…”

Trace functions of the parafermion vertex operator algebras

“…We review the modular transformation formula of the theta functions defined on vertex operator algebra given in [35] and [31] for studying the modular invariance of trace functions for the parafermion vertex operator algebras.…”

“…Then the abelian Lie algebra h acts on M i semsimply for all i. Following [35], we define the generalized theta functions as…”

“…for u, v ∈ h. The bilinear form on V 1 used in [35] is the negative of the bilinear form used in this paper. So our T i (v, u, q) defined here is exactly the Z M i (v, u, q) in [35].…”

Fusion rules for the vertex operator algebra VL2A4

“…Then we have the following theorem [32,36,16,19]: Theorem 2.14. Let V be a rational, C 2 -cofinite vertex operator algebra of CFT type.…”