Abstract

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 ${ℝ}^d$ 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 ${ℝ}^d$. 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.

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.

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