On the Genus Two Free Energies for Semisimple Frobenius Manifolds

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.