A Cohomological Interpretation of the Migdal-Makeenko Equations

TL;DR

This paper provides a cohomological interpretation of the Migdal-Makeenko equations in large N Yang-Mills theory, revealing an anomaly as a cohomology element related to loop-space variable changes.

Contribution

It introduces a cohomological framework for understanding the Migdal-Makeenko equations, connecting anomalies to Lie algebra cohomology in loop-space.

Findings

Anomalous term in loop-space Yang-Mills equations is interpreted as a cohomology element.

The Lie algebra of loop substitutions is developed and related to the anomaly.

The invariance of the generating functional is characterized by the Migdal-Makeenko equations.

Abstract

The equations of motion of quantum Yang - Mills theory (in the planar `large N' limit), when formulated in Loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones i.e. the connection one form $A $, to the holonomy $U$. An infinite dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal-Makeenko equations are shown to be the condition for the invariance of the Yang-Mills generating functional $Z$ under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.