Heat Kernels, Plancherel Duality, and the Selberg Trace Formula

<span>The Selberg trace formula and the Migdal–Witten solution of two-dimensional Yang–Mills theory are usually presented as belonging to unrelated fields-automorphic spectral geometry on one hand, exactly solvable quantum field theory on the other. We show they are two instances of a single mechanism: the heat kernel of a Lie group, decomposed through the group's Plancherel theorem, glued according to a discrete combinatorial identification.</span> <div> <br><span>For Selberg, the group is the non-compact, the identification is a cocompact Fuchsian group, and the relevant duality is the Harish-Chandra Plancherel formula for the principal series, whose spectral density is exactly the weight appearing in the trace formula's identity term. For Migdal–Witten two-dimensional Yang–Mills, the group is a compact gauge group (e.g.), the identification is the cell structure of agenus-surface, and the relevant duality is the Peter–Weyl theorem, whose finite-dimensional irreducible representations and their dimensions play the role of the principal series and its Plancherel density. We derive both trace formulas side by side from this common ancestor, make the dictionary between them precise (Section 4, Table1), and identify a further resonance: on both sides the "geometric" dual data is indexed by conjugacy classes of a group acting by holonomy -the Fuchsian group for Selberg, and (through Gross–Taylor duality and the Frobenius formula) the symmetric group via Hurwitz theory for large-Yang–Mills. Every non-trivial identity used is checked numerically: an exact (machine-precision) finite-group verification of the handle-gluing formula, a Monte Carlo verification of the heat-kernel semigroup law on, asymbolic derivation of the Douglas–Kazakov critical area, and a 25-digit numerical confirmation that the Harish-Chandra spectral integral reproduces McKean's closed-form hyperbolic heat kernel. We are explicit throughout about the limits of the correspondence: two-dimensional Yang–Mills has no propagating gluon degrees of freedom, so nothing here bears on confinement or the four-dimensional mass gap. The contribution is a rigorous, checkable unification of methods and a pedagogical bridge between two literatures that rarely cite each other.</span> </div>