Residue formulas for prepotentials, instanton expansions and conformal blocks

We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory con- densates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also the differential equations, following from the residue formulas, including the WDVV equations, proven to be valid for the SU(2) quiver gauge theories. As a particular example we consider the constrained conformal quiver gauge theory, corresponding to the Zamolodchikov conformal blocks by 4d/2d duality. In this case part of the found differential equations turn into nontrivial relations for the period matrices of hyperelliptic curves.