Extended Seiberg-Witten Theory and Integrable Hierarchy

TL;DR

This paper demonstrates that the prepotential in certain N=2 super-Yang-Mills theories is a tau-function of the quasiclassical Toda hierarchy, linking integrable systems with supersymmetric gauge theories and string theory.

Contribution

It extends Seiberg-Witten theory by connecting it to integrable hierarchies, explicitly solving the hierarchy for specific theories, and clarifying the geometric and matrix model origins.

Findings

Prepotential is a tau-function of the quasiclassical Toda hierarchy.

Explicit solutions for the hierarchy in specific theories are provided.

The Seiberg-Witten curve deformation by Toda flows is characterized.

Abstract

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 CP^1 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.