GKS-II-Type Correlation Inequalities for SU(2) Lattice Gauge Theory

We prove GKS-II-type correlation inequalities for SU(2) lattice gauge theory with the Wilson action in three geometries. (i) For two-dimensional YangMills theory on any closed orientable cellular surface of genus g, the covariance of two distinct plaquette variables equals the variance of an explicit function of the representation label under a positive probability measure: Cov(p,q) = Varw(φ) ≥ 0, φ(j) = aj−1/2(β) + aj+1/2(β) /aj(β). (ii) For the 1 × 1 × 2 box (two unit cubes sharing a face), Cov(p1,p2) ≥ 0 for all β > 0, for plaquettes p1,p2 lying in dierent cubes. The proof combines an Amos-type bound for ratios of modied Bessel functions with total positivity of the kernel I|n−m|(β)− In+m(β) = [eβJ]nm, J the free Jacobi matrix with Dirichlet boundary condition. (iii) For the 2 × 2 × 1 slab and two exterior plaquettes p0,p13 lying in diagonally opposite cubes, the covariance vanishes identically in the theory restricted to those two cubes (gauge xing on their single shared link), and in the full slab Cov(p0,p13) = 2−23β12 + O(β14), so the inequality holds strictly for 0 < β < β0. We also state a total-positivity conjecture for the higher-power fusion kernels that arise for further observables, supported by 547,560 high-precision checks. To our knowledge these are the rst rigorous GKS-II-type inequalities at non-zero coupling for a non- abelian lattice gauge theory; they are consistent with the general conjecture of Abdesselam and collaborators on correlation inequalities for non-abelian lattice gauge theories.