Universal dualities for Wilson loops in lattice Yang-Mills

Abstract

We identify a universal finite-$N$ structure underlying Wilson loop expectations in lattice Yang--Mills, in any dimension $d\ge 2$, for gauge group $\mathrm{U}(N)$, and for arbitrary smooth central plaquette actions. The starting point is a state-sum expansion in plaquette labels by irreducible representations, in which each term factorizes into an action-dependent spectral weight and an action-independent topological coefficient. We then analyze these coefficients in three exact ways: as a gauge/string expansion over decorated spanning surfaces, as a local spin-foam/channel model on the dual incidence graph, and as a universal finite-$N$ master loop equation that closes on the coefficient side. As a consequence, several recent Wilson-action results are recovered as specializations of our broader action-agnostic framework. In a companion work, this framework is used to construct the higher-dimensional heat-kernel master field at strong coupling and to prove an area-law bound.

arXiv commentV2: 60 pages, 11 figures. The local spin-foam/channel representation has been reformulated using edgewise Haar projections and scalarized local channels. The companion application to the heat-kernel master field is now included, and numerous proofs and expository points have been revised. V1: 59 pages, 11 figures. Comments welcome!