Free Fermion and Seiberg-Witten Differential in Random Plane Partitions

TL;DR

This paper explores the semi-classical wave functions of fermions in a random plane partition model related to five-dimensional supersymmetric gauge theories, revealing connections to Seiberg-Witten theory and RG flow.

Contribution

It demonstrates how fermionic wave functions in a random plane partition model relate to Seiberg-Witten curves and different limits, providing a new geometric perspective.

Findings

Wave functions become WKB type at thermodynamic limit

Semi-classical wave functions on Seiberg-Witten curve in 4D limit

Fermions off the main diagonal relate via RG flow

Abstract

A model of random plane partitions which describes five-dimensional $\mathcal{N}=1$ supersymmetric SU(N) Yang-Mills is studied. We compute the wave functions of fermions in this statistical model and investigate their thermodynamic limits or the semi-classical behaviors. These become of the WKB type at the thermodynamic limit. When the fermions are located at the main diagonal of the plane partition, their semi-classical wave functions are obtained in a universal form. We further show that by taking the four-dimensional limit the semi-classical wave functions turn to live on the Seiberg-Witten curve and that the classical action becomes precisely the integral of the Seiberg-Witten differential. When the fermions are located away from the main diagonal, the semi-classical wave functions depend on another continuous parameter. It is argued that they are related with the wave functions at the main diagonal by the renormalization group flow of the underlying gauge theory.