Extended $5d$ Seiberg-Witten Theory and Melting Crystal

TL;DR

This paper extends 5d Seiberg-Witten theory by connecting correlation functions of loop operators to melting crystal models, revealing integrable structures and geometric limits in supersymmetric gauge theories.

Contribution

It introduces a novel link between 5d supersymmetric Yang-Mills theory, melting crystal models, and integrable hierarchies, providing new computational and geometric insights.

Findings

Correlation functions expressed as melting crystal partition functions.

Identification of a Seiberg-Witten curve from the thermodynamic limit.

Connection to 1-Toda hierarchy via tau functions.

Abstract

We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills in the $\Omega$ background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a $\tau$ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential.