Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

TL;DR

This paper develops formulas for the partition and n-point functions of vertex operator algebras on genus two Riemann surfaces, analyzing their modular properties and deriving identities for specific free bosonic theories.

Contribution

It provides explicit formulas for genus two partition functions and n-point functions for free bosonic VOAs, extending the understanding of their modular and geometric properties.

Findings

Closed formulas for genus two partition functions for Heisenberg VOAs.

Holomorphicity of the partition function in sewing parameters.

Genus two Ward identity for the Virasoro vector.

Abstract

We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.