The $(2,5)$ minimal model on genus two surfaces

TL;DR

This paper studies the partition function of the $(2,5)$ minimal model on genus two surfaces, generalizing Rogers-Ramanujan functions, and explores their modular properties and expansions in terms of modular forms, with applications to conformal field theory.

Contribution

It introduces a new framework for understanding the genus two partition function of the $(2,5)$ minimal model, including explicit expansions and differential equations for key functions.

Findings

Partition function expressed as a 5-tuple of functions with modular transformation properties.

Explicit expansions around conical singularities using modular forms.

Derived a third-order ODE for the two-point function in non-vacuum representations.

Abstract

In the $(2,5)$ minimal model, the partition function for genus $g=2$ Riemann surfaces is given by a $5$-tuple of functions with appropriate transformation under the mapping class group. These functions generalise the two Rogers-Ramanujan functions for the torus. Their expansions around a locus of surfaces with conical singularities in the interior of the $g=2$ Siegel upper half plane are obtained in terms of standard modular forms. The dependence on the metric is controlled by a canonical choice of flat surface metrics. In the alternative case where a handle of the $g=2$ surface is pinched, our method requires knowledge of the two-point function of the fundamental lowest-weight vector in the non-vacuum representation of the Virasoro algebra, for which we derive a $3$\ts{rd} order ODE. In order to make the paper more accessible to mathematicians, the exposition includes a short introduction to conformal field theory on Riemann surfaces, which may be of independent interest.