Heat Kernels, Trace Formulae, and Plancherel Correspondences in Solvable Two-Dimensional Theories: A Unified Monograph in Six Papers

This monograph develops a unified, self-contained account of heat kernels, trace formulae, and Plancherel measures across a family of exactly solvable two-dimensional and finite theories. The organizing object is the heat kernel on a Lie group, expanded spectrally over the unitary dual with Gaussian weight exp(−t·C₂(R)/2). The central claim, developed across six integrated papers, is that three constructions — (i) the partition and Wilson-loop observables of two-dimensional Yang–Mills theory on an arbitrary surface, (ii) the spectral side of the Selberg trace formula on a locally symmetric space, and (iii) the gluing data of a two-dimensional topological field theory — are governed by one convolution semigroup and one resolution of the identity, related by explicit, non-conjectural maps rather than by analogy. The six chapters treat, in turn: heat-kernel correspondences across solvable 2D theories; an explicit dictionary between compact (Peter–Weyl, dim R) and non-compact (Harish–Chandra–Plancherel, |c(λ)|⁻²) Plancherel measures, with the ρ-shift identified as the analytic signature of curvature; the geodesic–holonomy correspondence and its extension beyond PSL(2,R) to higher-rank groups (geodesics → closed flats, scalar length → Weyl-chamber vector); numerical verification of the resulting trace-kernel identities for U(1), SU(2), and PSL(2,R), including a machine-precision check of the heat-kernel gluing law K_s∗K_t = K_{s+t}; finite-group analogues in which every sum is finite and every identity is exactly checkable, with the S_n case linked to Hurwitz numbers; and a reformulation of the heat-kernel semigroup as the cylinder (gluing) operator of a 2D topological field theory whose Frobenius algebra is the center of the group algebra. Appendices collect the special-function and representation-theoretic background, the numerical protocols, and a master table of correspondences; a glossary fixes a single notation used throughout, and one consolidated bibliography (Selberg, Harish-Chandra, Migdal, Rusakov, Witten, Gross–Taylor, McKean, Gindikin–Karpelevich, Hurwitz, HCIZ, and others) serves all six papers. Scope note: all results are theorems of classical harmonic analysis or exact finite-arithmetic statements, valid because the theories are two-dimensional or finite. The work makes no claim about four-dimensional Yang–Mills theory, quark confinement, or the Yang–Mills mass gap; the term "Selberg–Yang–Mills correspondence" denotes only the shared representation-theoretic structure (characters/spherical functions, Casimirs, heat kernels) of the Selberg trace formula and two-dimensional gauge theory. # KEYWORDSheat kernel, Plancherel duality, Selberg trace formula, Yang–Mills theory, representation theory, spectral geometry, hyperbolic geometry, Lie groups, BF theory, Chern–Simons theory, random matrix models, Hurwitz theory, geodesic spectrum, Casimir, spherical functions, conjugacy classes, numerical harmonic analysis, group theory, spectral decomposition, gluing rules # SUBJECTS- Mathematical Physics - Representation Theory - Spectral Geometry - Lie Groups and Harmonic Analysis - Topological Field Theory - Hyperbolic Geometry - Computational Mathematics - Gauge Theory (2D) - Automorphic Forms - Random Matrix Theory - Combinatorics of Finite Groups Contact: [email protected]