Mathematics

We represent the genus two free energy of an arbitrary semisimple Frobenius\\r\\nmanifold as a sum of contributions associated with dual graphs of certain\\r\\nstable algebraic curves of genus two plus the so-called \\\"genus two G-function\\\".\\r\\nConjecturally the genus two G-function vanishes for a series of important\\r\\nexamples of Frobenius manifolds associated with simple singularities as well as\\r\\nfor ${\\\\bf P}^1$-orbifolds with positive Euler characteristics. We explain the\\r\\nreasons for such Conjecture and prove it in certain particular cases.