Seiberg-Witten theory and matrix models

TL;DR

This work provides a microscopic matrix-integral framework for Seiberg-Witten theory in four and five dimensions by representing Nekrasov partition functions as matrix-model integrals whose spectral curves reproduce the Seiberg-Witten curves. In 4d, a 1-matrix model with a multi-minima potential $V^{4d}(x)$ captures the $SU(n)$ SW data, while in 5d a deformed Vandermonde and a dilogarithmic potential $V^{5d}(u)$ yield a trig (q-deformed) spectral curve, closely related to topological string theory on non-compact toric Calabi-Yau manifolds. The construction unifies several known matrix models (e.g., Eguchi-Yang, line bundles over $\mathbb{P}^1$, lens-space Chern-Simons) as limits of the general model, and connects geometric engineering, mirror symmetry, and topological strings through the shared spectral curve data. The results offer a concrete path to higher-genus amplitudes via the Eynard-Orantin formalism and suggest avenues to explore non-perturbative effects and OSV-type questions within a unified matrix-model framework.

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.

Figures (5)

• Figure 1:
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5: