Symplectic Capacity and the Strong-Coupling Spectrum of Lattice Yang-Mills: Capacity as an Integrality Certificate, and a Symplectic Lichnerowicz Conjecture

We connect the Gromov symplectic capacity of coadjoint orbits of compact Lie groups to the spectral gap of lattice Yang–Mills. For any compact simple gauge group, symplectic reduction of the lattice phase space contains coadjoint-orbit Kähler factors whose Gromov capacity is finite, positive, and coupling-independent; their integrality, through geometric quantization, fixes the discreteness of the electric spectrum, with the strong-coupling gap recovered as the Kogut–Susskind Casimir. We do not address the continuum limit, where the mass-gap problem resides. Separately, we pose a symplectic Lichnerowicz inequality for Fano Kähler–Einstein manifolds, explicitly decoupled from the mass-gap claim. Keywords: lattice gauge theory, Yang–Mills mass gap, symplectic capacity, geometric quantization, coadjoint orbits, spectral gap, Kähler–Einstein manifolds, math-ph, hep-lat, math.SG