Three proofs of the Makeenko-Migdal equation for Yang-Mills theory on the plane

TL;DR

This paper presents three simplified, local proofs of the Makeenko-Migdal equation for the Yang-Mills measure on the plane, improving upon earlier more complex proofs and with potential extensions to compact surfaces.

Contribution

It introduces three new, simpler proofs of the Makeenko-Migdal equation using edge and loop variables, emphasizing their local nature.

Findings

Proofs are significantly simpler than previous ones.

Two proofs can be adapted to compact surfaces.

Proofs involve only derivatives near crossings.

Abstract

We give three short proofs of the Makeenko-Migdal equation for the Yang-Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by T. L\'evy and by A. Dahlqvist. In particular, our proofs are "local" in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with F. Gabriel, we will show that two of our proofs can be adapted to the case of Yang-Mills theory on a compact surface.