Equivariant measure localization and spectral geometry: a rigorous proof programme for the Yang-Mills mass gap

Abstract We present a detailed proof programme for the existence of a mass gap ∆ > 0 in pure YangMills gauge theory with structure group G = SU(N), N ≥ 2, on a compact four-dimensional Riemannian manifold (X, g). The argument proceeds in three stages. Stage I. We establish the rigorous wellposedness of the YangMills functional measure in the sense of cylindrical measures on the orbit space B = A/G (Measure Well-Posedness Theorem, Ÿ Stage II. Using an innite-dimensional analogue of the DuistermaatHeckman theorem in the form developed for Q-topological theories, we prove that in the infrared limit the entire probability mass concentrates on the disjoint union of nite-dimensional instanton moduli spaces Fk∈Z Mk (Infrared Localisation Theorem, Ÿ Stage III. Applying Witten's deformation to the LaplaceBeltrami operator and establishing the strict positivity of the Cheeger isoperimetric constant h(B) > 0 via an explicit sphaleron barrier estimate, we derive the sharp inequality ∆ ≥ h(B)2/4 > 0 (Mass Gap Theorem, Ÿ 5). At each step we state explicitly the assumptions that distinguish rigorous results from physically motivated conjectures, and indicate pathways for elevating those conjectures to theorems.