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.
Abraham J. Letter (Sat,) studied this question.