Proof of a Conjecture on the Genus Two Free Energy Associated to the A_n Singularity

TL;DR

This paper proves a conjecture that the genus two G-function vanishes for Frobenius manifolds related to A-type singularities, simplifying the understanding of their genus two free energy structure.

Contribution

It establishes the vanishing of the genus two G-function specifically for Frobenius manifolds associated with A-type singularities, confirming a conjecture in the field.

Findings

Proves the vanishing of the genus two G-function for A_n singularities.

Simplifies the expression for genus two free energy in these cases.

Supports the broader conjecture for a class of Frobenius manifolds.

Abstract

In a recent paper [8], it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two G-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two G-function vanishes. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.