On Microscopic Origin of Integrability in Seiberg-Witten Theory

TL;DR

This work investigates the microscopic origin of integrability in Seiberg-Witten theory by leveraging Nekrasov’s instanton counting and a free-fermion/bosonization framework to identify the Seiberg-Witten prepotential $\mathcal{F}$ as a quasiclassical tau-function of a dispersionless Toda hierarchy. It develops a variational formulation for the quasiclassical free energy and analyzes both Abelian and non-Abelian cases, revealing how UV perturbations deform the spectral curve and how higher flows generate explicit, computable structures via elliptic uniformization (notably for $U(2)$). The results connect instanton counting with integrable hierarchies, provide explicit residue formulas for derivatives of $\mathcal{F}$, and establish a concrete route to explicit quasiclassical solutions on hyperelliptic curves, while highlighting open questions about all-orders exactness and matter content.

Abstract

We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of hep-th/0612019, as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of the spectral curve.

Figures (1)

• Figure 1: Prepotential, as a sum of all connected tree diagrams in the bosonic cubic field theory. Each vertex is weighted by $t_2$ and each pair of external legs -- by $e^{{\bf t}"(a)}$. The depicted diagramms correspond literally to the contribution of truncated "bosonic BCS model", with the dashed lines being the $\langle AA^\dagger\rangle$-"propagators".