Genus Two Zhu Theory for Vertex Operator Algebras

Abstract

We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion where we express an arbitrary genus two n-point correlation function in terms of (n − 1)-point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half plane for holomorphic 1-differentials, the normalised bidifferential of the second kind, the projective connection and the Heisenberg partition function. We prove that the holomorphic mapping from the sewing parameter domain to the Siegel upper half plane is injective but not surjective. We also demonstrate that genus two differential equations arising from Virasoro singular vectors have holomorphic coefficients.

Genus Two Virasoro Correlation Functions for Vertex Operator Algebras

“…We first show that genus two n-point correlation functions for n Virasoro vectors are generating functions for the correlation functions for all Virasoro vacuum descendants in a similar fashion to the genus zero and one cases [HT1]. These generating functions satisfy genus two Ward identities (derived from genus two Zhu recursion) which involves genus two generalised Weierstrass functions related to a global (2,1)-bidifferential Ψ(x, y) holomorphic for x = y [GT1]. We also describe some analytic differential equations, which also involve Ψ(x, y), for the genus two bidifferential ω(x, y), normalised holomorphic 1-differentials ν 1 (x), ν 2 (x) and the projective connection s(x).…”

“…We define the genus two partition function and n-point correlation function for a VOA based on the sewing scheme for S (2) constructed from two tori S 1 and S 2 [MT3,GT1]. The genus two partition function for V of strong CFT-type is defined by…”

“…Correlation functions for VOAs on a genus two Riemann surface have been defined, and in some cases calculated based on an explicit sewing procedure for sewing two tori together [MT2,MT3]. We recently described a general Zhu recursion formula for genus two n-point correlation functions which gives rise to new genus two Ward identities when applied to Virasoro vector n-point functions [GT1]. The purpose of this paper is to describe all genus two correlation functions for Virasoro descendants of the vacuum vector in terms of explicit generating functions.…”

Cohomology of modular form connections on complex curves

“…Chain complex spaces of n-variable modular forms. In this section we introduce the chain complex spaces for modular functions on complex curves [EZ,Zag,Zhu,Miy,Miy1,MTZ,GT,TW]. Mark n points p n " pp 1 , .…”

Reduction cohomology of Riemann surfaces

“…Recall that we consider genus g Riemann surfaces resulting from combinations of sewing procedures of [Y]. Accordingly, due to [MT,TZ,TZ1,TZ2,GT,TW], corresponding genus g n-point functions are obtained coherently by combining lower genus functions. Then, relations among n-point functions of various genera appear.…”