A Functional Integral Approaches to the Makeenko-Migdal Equations

Makeenko and Migdal (1979) gave heuristic identities involving the expectation of the product of two Wilson loop functionals associated to splitting a single loop at a self-intersection point. Kazakov and I. K. Kostov (1980) reformulated the Makeenko--Migdal equations in the plane case into a form which made rigorous sense. Nevertheless, the first rigorous proof of these equations (and their generalizations) was not given until the fundamental paper of T. L\'{e}vy (2011). Subsequently Driver, Kemp, and Hall (2017) gave a simplified proof of L\'{e}vy's result and then with F. Gabriel (2017) we showed that these simplified proofs extend to the Yang-Mills measure over arbitrary compact surfaces. All of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts. The goal of this article is to give a rigorous functional integral proof of the Makeenko--Migdal equations guided by the original heuristic machinery invented by Makeenko and Migdal. Although this stochastic proof is technically more difficult, it is conceptually clearer and explains "why" the Makeenko--Migdal equations are true. It is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing Yang-Mill's fields.