Wilson loop expectations as sums over surfaces on the plane

Although lattice Yang-Mills theory on finite subgraphs of ${\mathbb{Z}}^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on ${\mathbb{R}}^d$ is a major open problem when $d\ge 3$. Such a theory should in some sense assign a Wilson loop expectation to each suitable finite collection $\mathcal{L}$ of loops in ${\mathbb{R}}^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal{L}$. 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.