Status of the Kazakov--Migdal Model

TL;DR

This paper reviews the status and recent developments of the Kazakov--Migdal Model, highlighting its limitations in modeling QCD and exploring alternative solvable matrix models using loop equations.

Contribution

It provides a critical assessment of the Kazakov--Migdal Model's inability to induce QCD and introduces a new exactly solvable Penner-like matrix model.

Findings

Kazakov--Migdal Model does not induce QCD

A new exactly solvable matrix model with logarithmic singularities is presented

Loop equations are used to analyze complex matrix models

Abstract

In this talk I discuss both the present status and some recent work on the Kazakov--Migdal Model which was originally proposed as a soluble, large $N$ realization of QCD. After a brief description of the model and a discussion of its solubility in the large $N$ limit I discuss several of the serious problems with the model which lead to the conclusion that it does {\it not} induce QCD. The model is nonetheless a very interesting example of a Gauge Theory and it is related to some very interesting Matrix Models. I then outline a technique \REF\dmsxyz{\dms}\refend which uses ``Loop Equations'' for solving such models. A Penner--like model is then discussed with two logarithmic singularities. This model is distinguished by the fact that it is exactly and explicitly soluble in spite of the fact that it is not Gaussian. It is shown how to analyze this model using both a technical approach and from a more physical point of view.