Wilson loop expectations as sums over surfaces on the plane

2023/05/03 by Park, Minjae, Pfeffer, Joshua, Sheffield, Scott +1 · 3 citations
#58J65 #60D05 (Primary) #60H25 #70S15 #81T13 #81T35 #82B41 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2305.02306

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 \mathbb R^d 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 \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.

Cited by

• Makeenko-Migdal equations for lattice Yang-Mills-Higgs
• The central heat trace on large compact classical groups
• Two-dimensional Yang-Mills theory via integrable probability

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