Although lattice Yang-Mills theory is easy to rigorously define, the construction of a satisfactory continuum theory is a major open problem in dimension d ≥ 3. Such a theory should assign a Wilson loop expectation to each suitable collection L of loops in d-dimensional space. One classical approach is to try to represent this expectation as a sum over surfaces having L as their boundaries. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.
I will introduce the subject and show how to make sense of Yang-Mills integrals as surface sums for d=2, where the continuum theory is already understood. This perspective leads to alternative proofs of the Makeenko-Migdal equation and the Gross-Taylor expansion. The presentation is based on a joint work with Minjae Park (Chicago), Joshua Pfeffer (Columbia) and Pu Yu (MIT).