Proof of a conjecture on the genus two free energy associated to the AnAn singularity

In a recent paper Dubrovin et al. (1998), 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 GG-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two GG-functions vanish. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.