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We study various properties of a nonperturbative partition function which canbe associated to any spectral curve. When the spectral curve arises from amatrix model, this nonperturbative partition function is given by a sum ofmatrix integrals over all possible filling fractions, and includes all themulti-instanton corrections to the perturbative 1/N expansion. We show that thenonperturbative partition function, which is manifestly holomorphic, is alsomodular and background independent: it transforms as the partition function ofa twisted fermion on the spectral curve. Therefore, modularity is restored bynonperturbative corrections. We also show that this nonperturbative partitionfunction obeys the Hirota equation and provides a natural nonperturbativecompletion for topological string theory on local Calabi-Yau threefolds.

A holomorphic and background independent partition function for matrix models and topological strings