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 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.

25 pages. v3: misprints corrected,references added

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• Initial-boundary value problem and long-time asymptotics for the Kundu--Eckhaus equation on the half-line
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions
• An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times 4$ Lax pair
• Riemann-Hilbert approach for a mixed coupled nonlinear Schrödinger system and its soliton solutions