Genus Two Meromorphic Conformal Field Theory

TL;DR

This paper constructs the genus two partition function for meromorphic bosonic conformal field theories, revealing connections to Siegel modular forms and providing explicit formulas for various theories including the Moonshine Module.

Contribution

It introduces a sewing procedure to compute genus two partition functions and relates them to Siegel modular forms, advancing understanding of higher-genus conformal field theory invariants.

Findings

Partition functions expressed via sewing of genus one tori

Genus two partition function for self-dual theories linked to Siegel modular forms

Explicit formulas for Moonshine Module and lattice theories

Abstract

We construct the genus two (or two loop) partition function for meromorphic bosonic conformal field theories. We use a sewing procedure involving two genus one tori by exploiting an explicit relationship between the genus two period matrix and pinching modular parameters. We obtain expressions for the partition function for the chiral bosonic string, even rank lattice theories and self-dual meromorphic conformal field theories including the Moonshine Module. In particular, we find that for self-dual theories with central charge 24, the genus two partition function multiplied by a universal holomorphic function of the moduli is given by a meromorphic Siegel modular form of weight 2 where this universal function includes ghost contributions. We also discuss a novel expansion for certain Siegel modular forms.