On the Torus Degeneration of the Genus Two Partition Function

TL;DR

This paper studies how the genus two partition function of a vertex operator algebra degenerates into genus one functions when one torus pinches, revealing a universal factor related to the central charge.

Contribution

It provides an explicit analysis of the degeneration limit of genus two partition functions, connecting them to genus one functions with a universal multiplicative factor.

Findings

Genus two partition function reduces to genus one in the degeneration limit.

The universal factor depends only on the central charge and is independent of the VOA.

The result clarifies the behavior of VOAs on degenerating Riemann surfaces.

Abstract

We consider the partition function of a general vertex operator algebra $V$ on a genus two Riemann surface formed by sewing together two tori. We consider the non-trivial degeneration limit where one torus is pinched down to a Riemann sphere and show that the genus one partition function on the degenerate torus is recovered up to an explicit universal $V$-independent multiplicative factor raised to the power of the central charge.