On the Genus Two Free Energies for Semisimple Frobenius Manifolds

TL;DR

This paper expresses the genus two free energy of semisimple Frobenius manifolds as a sum over dual graphs and a G-function, conjecturing the G-function's vanishing in key cases and proving it in specific instances.

Contribution

It introduces a new representation of genus two free energy involving dual graphs and the G-function, and proves the G-function's vanishing in particular cases.

Findings

Representation of genus two free energy as sum over dual graphs and G-function

Conjecture that the G-function vanishes for important Frobenius manifolds

Proof of the G-function's vanishing in specific cases

Abstract

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold 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". Conjecturally the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for ${\bf P}^1$-orbifolds with positive Euler characteristics. We explain the reasons for such Conjecture and prove it in certain particular cases.