Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory

TL;DR

The paper studies the exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory within the Whitham hierarchy framework, identifying it with a homogeneous solution of a Whitham hierarchy (the Whitham-Toda hierarchy). The hierarchy describes modulation of quasi-periodic solutions of the generalized Toda lattice associated with hyperelliptic curves over the quantum moduli space, and the holomorphic pre-potential $F$ is related to the $tau$-function of this generalized Toda lattice hierarchy. A meromorphic differential $dS$ on the hyperelliptic curve is constructed and decomposed to connect to the low-energy data via period integrals on beta cycles, establishing relations for the derivatives of $F$ with respect to the moduli and flows. By introducing slow-time flows and a Baker-Akhiezer function, the modulation yields Virasoro-type constraints on $F$, suggesting a potential two-topological-string interpretation of the $N=2$ theory.

Abstract

The exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasi-periodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the $τ$ function of the (generalized) Toda lattice hierarchy is also clarified.