Seiberg–Witten theory as a Fermi gas

Abstract

We explore a new connection between Seiberg–Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang–Mills theory in a self-dual $$\Omega $$ background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painleve $$\mathrm{III}_3$$ $$\tau $$ function. In addition, we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local $${\mathbb P}^1 \times {\mathbb P}^1$$ geometry.