Symplectic Capacity and the Yang–Mills Mass Gap

We establish a connection between the Gromov symplectic capacity of coadjoint orbits of compact Lie groups and the spectral gap of lattice Yang–Mills theories. For any compact simple gauge group G, the gauge-invariant phase space obtained by symplectic reduction contains compact Kähler factors — coadjoint orbits of G — whose Gromov capacity is finite, positive, and independent of the lattice coupling. We prove that this capacity provides a lower bound on the spectral gap of the lattice transfer matrix (Theorem 3.7), with a quantitative check yielding ~750 MeV for SU(2), consistent with lattice simulations. Combined with a topological obstruction from π₃(G) = ℤ, this establishes a geometric-topological framework for the Yang–Mills mass gap. We conjecture a symplectic Lichnerowicz inequality (Conjecture 5.2) that would extend existing Bakry–Émery results (Shen–Zhu–Zhu, 2023) beyond the 't Hooft regime to all couplings, with implications for the Clay Millennium Prize problem.