The planar limit of $\mathcal{N}=2$ superconformal quiver theories

TL;DR

This paper computes the planar limit of free energy and Wilson loop expectation values in four-dimensional ${ m extbf{N}=2}$ superconformal quiver theories, revealing a sum-over-trees structure and a connection to generalized Ising models.

Contribution

It introduces a method to analyze ${ m extbf{N}=2}$ quiver theories using multi-matrix models and tree graph sums, extending previous results for simple gauge groups.

Findings

Results expressed as sums over tree graphs.

Each tree's contribution interpreted as a generalized Ising model partition function.

Conjecture that these partition functions satisfy the Lee-Yang property.

Abstract

We compute the planar limit of both the free energy and the expectation value of the $1/2$ BPS Wilson loop for four dimensional ${\cal N}=2$ superconformal quiver theories, with a product of SU($N$)s as gauge group and bi-fundamental matter. Supersymmetric localization reduces the problem to a multi-matrix model, that we rewrite in the zero-instanton sector as an effective action involving an infinite number of double-trace terms, determined by the relevant extended Cartan matrix. We find that the results, as in the case of $\mathcal{N}=2$ SCFTs with a simple gauge group, can be written as sums over tree graphs. For the $\widehat{A_1}$ case, we find that the contribution of each tree can be interpreted as the partition function of a generalized Ising model defined on the tree; we conjecture that the partition functions of these models defined on trees satisfy the Lee-Yang property, i.e. all their zeros lie on the unit circle.