Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szegö kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.


Vertex Operator Super Algebras
In this paper (based on several conference talks of the author) we review our recent results [1][2][3][4][5] on construction and computation of correlation functions of vertex operator superalgebras with a formal parameter associated to local coordinates on a self-sewn Riemann surface of genus g which forms a genus 1 g  surface. In particular, we review result presented in the papers [1][2][3][4][5] accomplished in collaboration with M. P. Tuite (National University of Ireland, Galway, Ireland).

Twisted Modules
We next define the notion of a twisted -module [8,11]. Let V g be a V -automorphism g , i.e., a linear map preserving and 1 We assume that can be decomposed into the vector space of ( End g W )-valued formal series in with arbitrary complex powers of . For The g -twisted vertex operators satisfy the twisted Jacobi identity:

Creative Intertwining Operators
We define the notion of creative intertwining operators in [3]. Suppose we have a VOA V with a V -module The operators with some extra cocycle structure satisfy a natural extension from rational to complex parameters of the notion of a Generalized VOA as described by Dong and Lepowsky [7,12]. We then prove in [3].

Invariant Form for the Extended Heisenberg Algebra
The definitions of invariant forms [8,13] for a VOSA and its g -twisted modules were given by Scheithauer [14] and in [2]   , and normalized differential of the third kind are g by

Torus Self-Sewing to Form a Genus Two
ribe procedures of sewing Riemann surfaces [17]. Consider a self-sewing of the oriented torus

Riemann Surface
In [1] we desc Define the annuli ,

Elliptic Functions Function
The Weierstrass

Eisenstein Series
The Eisenstein series

Twisted Elliptic Functions
1 denote a pair of modulus one conver of the do ges absolutely and uniformly on compact subsets main < < z q q Lemma 1 (Mason-Tuite multipliers  and  respectively.

sted Weiers ass Functions
Define th ndard action of e modular group for wit

Twisted Elliptic Functions
ar,

he Prime Form
There exists a (nonsingular and odd) character be a holomorphic 1-form, and let   We define the prime form The prime form is anti-sy and a holomorphic differential form of weight We have and is a meromorphic Finall be the m e y, we descri odular invarianc of the Szegö kernel under the symplectic group

Modular Properties of the Szegö Kernel
y, we describe the modula nvariance of the Szegö kernel under the symplectic group where we find [21] Finall r i  

Torus Szegö Kernel
On the tor   the Szegö kernel for     , 1,

A Determinant and the Period Matrix
Consider the infinite matrix

Theorem 9 (Tuite-Z)
a) The infinite matrix is non-vanishing and holomorphic on   .

Genus Two Szegö Kernel in the ρ -Formalism
It is convenient to define 1 1 , Then we prove [1] the following orphic in

Let
, for and integer We introduce the moments for with associated infinite matrix . We define also rder differential Defining   x y is given by

Graded
We define the genus one partition function for the VOSA by the supertrace

Genus One Fermionic One-Point Functions
Each orbifold 1-point function can found from a generalized Zhu reduction formulas as a determinant.

Genus One n-Point Functions for VOA
In general, we can define the genus one orbifold oint function for V n-p Every orbifold n-point function can be computed using generalized Zhu reduction formulas in terms of a determinant with entries arising from the basic 2-point function for ,     [19]. .

Zhu Reduction Formula
To reduce an -point function to a sum of point functions we need: The supertrace property , expansions for -functions:

General Genus One Fermionic n-Point Functions
The generating two-point function (for     , 1, where  is certain parity factor. Here M is the block associated to the torus intertwined point function.

Torus Intertwined Two-Point Function
The rank two free fermionic VOSA