Seiberg-Witten theory and matrix models

TL;DR

This paper develops matrix models that encode solutions to Seiberg-Witten theories in 4 and 5 dimensions, linking them to Nekrasov partition functions, spectral curves, and topological string theories on Calabi-Yau manifolds.

Contribution

It introduces a unified matrix model framework for Seiberg-Witten solutions, connecting various existing models and extending to topological string theory via geometric engineering.

Findings

Partition functions match Nekrasov functions

Spectral curves correspond to Seiberg-Witten curves

Unification of multiple matrix models

Abstract

We derive a family of matrix models which encode solutions to the Seiberg-Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg-Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi-Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi-Yang matrix model, matrix models for bundles over $P^1$, and Chern-Simons matrix models for lens spaces, which arise as various limits of our general result.