Extended Seiberg-Witten theory and integrable hierarchy

The prepotential of the effective ≤ 2 super-Yang-Mills theory, perturbed in the ultraviolet by the descendents ∫d⁴θ tr Φ^k+1 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, and we present its explicit solution. Our results generalise the limit shape analysis of Logan-Schepp and Vershik-Kerov, support the prior work [1], which established the equivalence of these ≤ 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.