Twisted Six Dimensional Gauge Theories on Tori, Matrix Models, and Integrable Systems

TL;DR

This paper explores the intricate relationships between six-dimensional gauge theories, matrix models, and integrable systems, revealing new geometric structures and integrability properties through advanced mathematical techniques.

Contribution

It introduces a novel matrix model on a torus with a genus 2 spectral curve and establishes a connection between the integrable system's phase space and the moduli space of Seiberg-Witten curves.

Findings

Matrix model with genus 2 spectral curve on a torus.

Explicit Poisson commuting Hamiltonians for the integrable system.

Equilibrium points correspond to maximally degenerated Seiberg-Witten curves.

Abstract

We use the Dijkgraaf-Vafa technique to study massive vacua of 6D SU(N) SYM theories on tori with R-symmetry twists. One finds a matrix model living on the compactification torus with a genus 2 spectral curve. The Jacobian of this curve is closely related to a twisted four torus T in which the Seiberg-Witten curves of the theory are embedded. We also analyze R-symmetry twists in a bundle with nontrivial first Chern class which yields intrinsically 6D SUSY breaking and a novel matrix integral whose eigenvalues float in a sea of background charge. Next we analyze the underlying integrable system of the theory, whose phase space we show to be a system of N-1 points on $T$. We write down an explicit set of Poisson commuting Hamiltonians for this system for arbitrary N and use them to prove that equilibrium configurations with respect to all Hamiltonians correspond to points in moduli space where the Seiberg-Witten curve maximally degenerates to genus 2, thereby recovering the matrix model spectral curve. We also write down a conjecture for a dual set of Poisson commuting variables which could shed light on a particle-like interpretation of the system.