Abstract
We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as the 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 ℙ1-orbifolds with positive Euler characteristics. We explain the reasons for the conjecture and prove it in particular cases.
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It also depends on the choice of the so-called calibration of the Frobenius manifold, i.e., on the choice of basis of horizontal sections of the deformed flat connection on M. See [5] for details.
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Dubrovin, B., Liu, SQ. & Zhang, Y. On the genus two free energies for semisimple Frobenius manifolds. Russ. J. Math. Phys. 19, 273–298 (2012). https://doi.org/10.1134/S1061920812030028
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DOI: https://doi.org/10.1134/S1061920812030028