Seiberg-Witten theory as a Fermi gas

TL;DR

This work establishes a precise 4d double-scaling limit in which the ghm spectral-determinant framework for toric Calabi–Yau geometries reduces to Seiberg–Witten theory in a self-dual $\Omega$-background, mediated by a Painlevé ${\mathrm{III}}_{3}$ tau function. The dual Nekrasov partition function becomes the grand partition function of an ideal one-dimensional Fermi gas, while the canonical Nekrasov function arises from an $O(2)$ matrix-model description that reproduces the SW prepotential and its gravitational corrections in the magnetic frame. The authors prove the ghm conjecture in this 4d regime by matching the spectral determinant with the tau-function, both on the topological-string side and in operator language (via Zamolodchikov-type kernels). This work thus unifies SW theory, Painlevé dynamics, topological strings, and quantum statistical systems, and opens paths to generalizations to theories with matter, higher rank, 5D, and lattice quantization.

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 III 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 P1xP1 geometry.