Extended Seiberg-Witten Theory and Integrable Hierarchy

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The prepotential of the effective N=2 super-Yang-Mills theory perturbed in the ultraviolet by the descendents of the single-trace chiral operators is shown to be a particular tau-function of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain. We present its explicit solutions. Our results generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support the prior work hep-th/0302191 which established the equivalence of these N=2 theories with the topological A string on CP¹ and clarify the origin of the Eguchi-Yang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical tau-function, show how the Seiberg-Witten curve is deformed by Toda flows, and fix the contact term ambiguity.