Abstract
We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates 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.
Highlights
General class of the quasiclassical tau-functions [9], which are well-known from long ago [10] to appear in the context of supersymmetric gauge theories
We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings
We are going to show in section, that equations (4.13) are immediately rewritten in the form of differential equations for effective couplings, which take the form of the Rauch relations, and can be implicitly solved via the Thomae formulas [45, 46]
Summary
We start from the definition of the SW system [2], assuming here the S-duality class of quiver gauge theories [1] with the k SU Nc(k) gauge groups. The connection ∇ on the moduli space, such that derivatives ∇ ∂ dS are holomorphic. G g δaiaDi = δ dS · dS is closed, so locally it is the differential of some function F. which covers Σ → Σ0 some curve Σ0 (which is often called UV or Gaiotto curve), whose moduli space can be parameterized by q = {q1, . We shall consider below the Zamolodchikov or constrained case, where the number of gauge theory condensates is constrained by certain conservation conditions (or vanishing of the masses of some light physical states), but the number of UV couplings remains intact, the reduced genus g < n. The set of the regular points on the cover Σ, where xdz has the simple poles with fixed residues (massive case). All mixed second derivatives are equal due to the RBR
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