Chatterjee’s analysis of three-dimensional lattice gauge theory [2, 3] establishes a confinementtype bound for any compact subgroup G ⊆ U(n) that contains the central circle {zI : |z| = 1}. The presence of this central U(1) is treated as a hypothesis on the gauge group. We observe, as a direct consequence of standard structure theory, that the hypothesis is automatically satisfied whenever G is the maximal compact subgroup of a non-compact simple real Lie group of Hermitian type: the central circle is then forced by the structure theory of the maximal compact subgroup. The same mechanism applies uniformly to the four classical Hermitian symmetric series (AIII, BDI, CI, DIII) (Corollary 2.6); the exceptional pairs EIII, EVII are covered by the same Helgason VIII.6–7 argument and are recorded in Remark 2.5. Specialising to G = SO(5)×SO(2) acting on the holomorphic tangent space of the type-IV5 Lie ball, we compute the weight of the intrinsic central circle explicitly and show it equals 1.
We then analyse a lattice Hermitian σ-model whose vertex variables take values in a compact arithmetic quotient DΓ of the bounded symmetric domain D = SO0(5, 2)/(SO(5) × SO(2)) of type IV5, with link weight given by the Bergman heat kernel Ht(σ, σ′). The heat-kernel choice makes reflection positivity manifest from semigroup factorisation. The transfer matrix is positive, self-adjoint, Hilbert–Schmidt, and irreducible, and the Krein–Rutman theorem yields a strictly positive spectral gap at every lattice spacing and every lattice volume. A Dobrushin uniqueness argument gives existence and uniqueness of the infinite-volume DLR Gibbs measure in a large-t (weak-coupling / high-temperature) regime t > t1 (OS axiom stability under the DLR limit follows from weak convergence of Schwinger functions and the linearity of the OS3 inequality; cf. Glimm–Jaffe §6.1), and lattice OS reconstruction produces a lattice OS-reconstructed Hilbert space carrying a self-adjoint transfermatrix Hamiltonian with hypercubic symmetry. Dobrushin uniqueness yields exponential decay of truncated correlations ([24, §IV.4], [25]); on the OS-reconstructed Hilbert space this transfers, via the K¨all´en–Lehmann representation, to a uniform-in-L lower bound on the lower edge of the spectral measure, yielding a strictly positive, but weak (Δ∞(t) → 0 as t → ∞), infinite-volume gap. The continuum limit and restoration of full Euclidean covariance are not established here and are left for future work. The Hermitian symmetric structure is load-bearing throughout: bounded-domain geometry gives e−Slink ≤1; parallel curvature ∇R ≡ 0 makes the Seeley–DeWitt coefficients of Ht field-independent constants, controlling uniform Hilbert–Schmidt and Dobrushin estimates. The argument uses only standard references (Helgason, Hua, Gilkey, Borel, Krein–Rutman, Dobrushin, L¨uscher, Osterwalder–Schrader, Osterwalder–Seiler, Glimm–Jaffe).
Publication Date: 2026-06-18