Macdonald's Identities and the Large N Limit of Y M₂ on the Cylinder

The purpose of this paper is to determine the large N asymptotics of the free energy F_N (a, U\A) of Y M₂ (two-dimensional Yang Mills theory) with gauge group G_N = SU(N) on a cylinder where a is a so-called principal element of type ρ. Mathematically F_N (U ₁, U₂\A) = 1/N² log H_GN (A/2N, U₁, U₂) where H_GN is the central heat kernel of G_N. We find that F_N(a_N, U_N\A) ∼ N/A Ξ(dθ, dσ) where Ξ is an explicit quadratic functional in the limit distribution dσ of eigenvalues of U_N, depending only on the integral geometry of SU(2). The factor of N contradicts some predictions in the physics literature on the large W limit of Y M₂ on the cylinder (due to Gross-Matytsin, Kazakov-Wynter, and others).