Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II

TL;DR

This paper develops formulas for genus two partition functions of free bosonic and lattice vertex operator algebras on Riemann surfaces formed by sewing a handle onto a torus, analyzing their modular properties and correlation functions.

Contribution

It provides explicit formulas for genus two partition functions and correlation functions, and studies their modular and degeneration properties, extending previous work on higher genus surfaces.

Findings

Partition functions are holomorphic in sewing parameters.

Genus two Heisenberg n-point functions are computed.

Partition functions are not compatible near a two-tori degeneration point.

Abstract

We continue our program to define and study $n$-point correlation functions for a vertex operator algebra $V$ on a higher genus compact Riemann surface obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras $V_L$. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. We also compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity. We compare our results with those obtained in the companion paper, when a pair of tori are sewn together, and show that the partition functions are not compatible in the neighborhood of a two-tori degeneration point. The \emph{normalized} partition functions of a lattice theory $V_L$ \emph{are} compatible, each being identified with the genus two theta function of $L$.