Kazakov-Migdal model on the Graph and Ihara Zeta Function

TL;DR

This paper introduces a Kazakov-Migdal model on graphs, linking it to the Ihara zeta function and Wilson loops, and provides exact large N solutions expressed through zeta functions.

Contribution

It establishes a novel connection between the Kazakov-Migdal model, Ihara zeta functions, and Wilson loops on graphs, with exact large N evaluations.

Findings

Partition function represented by an extended Ihara zeta function.

Duality allows finite N unitary matrix integrals.

Exact large N solution expressed as an infinite product of Ihara zeta functions.

Abstract

We propose the Kazakov-Migdal model on graphs and show that, when the parameters of this model are appropriately tuned, the partition function is represented by the unitary matrix integral of an extended Ihara zeta function, which has a series expansion by all non-collapsing Wilson loops with their lengths as weights. The partition function of the model is expressed in two different ways according to the order of integration. A specific unitary matrix integral can be performed at any finite $N$ thanks to this duality. We exactly evaluate the partition function of the parameter-tuned Kazakov-Migdal model on an arbitrary graph in the large $N$ limit and show that it is expressed by the infinite product of the Ihara zeta functions of the graph.