Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras II

TL;DR

This paper computes the genus two partition and correlation functions for a fermionic vertex operator superalgebra, revealing their modular properties and deriving a new identity for genus two theta series.

Contribution

It introduces explicit formulas for genus two partition and correlation functions of fermionic VOAs, including their modular behavior and a novel theta series identity.

Findings

Partition function proportional to an infinite determinant involving Szego kernel

Correlation functions expressed as finite determinants of genus two Szego kernels

Proven holomorphicity and modular properties of the functions

Abstract

We define and compute the continuous orbifold partition function and a generating function for all $n$-point correlation functions for the rank two free fermion vertex operator superalgebra on a genus two Riemann surface formed by self-sewing a torus. The partition function is proportional to an infinite dimensional determinant with entries arising from torus Szego kernel and the generating function is proportional to a finite determinant of genus two Szego kernels. These results follow from an explicit analysis of all torus $n$-point correlation functions for intertwiners of the irreducible modules of the Heisenberg vertex operator algebra. We prove that the partition and $n$-point correlation functions are holomorphic on a suitable domain and describe their modular properties. We also describe an identity for the genus two Riemann theta series analogous to the Jacobi triple product identity.