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

TL;DR

This paper develops genus two correlation functions for fermionic vertex operator superalgebras, providing explicit formulas, modular properties, and new identities, advancing the understanding of their structure on complex Riemann surfaces.

Contribution

It introduces a novel genus two correlation framework for fermionic vertex operator superalgebras, including explicit partition functions and modular analysis.

Findings

Derived a closed-form genus two partition function using Szeg"o kernels.

Proved the holomorphicity and modular properties of the partition function.

Established a new genus two Jacobi product identity for Riemann theta series.

Abstract

We define the partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szeg\"o kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all $n$-point functions in terms of a genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.