Wilson loop expectations as sums over surfaces on the plane

Although lattice Yang-Mills theory on finite subgraphs of Zᵈ is easy to rigorously define, the construction of a satisfactory continuum theory on Rᵈ is a major open problem when d ≥ 3. Such a theory should in some sense assign a Wilson loop expectation to each suitable finite collection L of loops in Rᵈ. One classical approach is to try to represent this expectation as a sum over surfaces with boundary L. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities. In this paper, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is more accessible. Applications include several new explicit calculations, a new combinatorial interpretation of the master field, and a new probabilistic proof of the Makeenko-Migdal equation.