Is the Two-Dimensional One-Component Plasma Exactly Solvable?

The model under consideration is the two-dimensional (2D) one-component plasma of point-like charged particles in a uniform neutralizing background, interacting through the logarithmic Coulomb interaction. Classical equilibrium statistical mechanics is studied by non-traditional means. The question of the potential integrability (exact solvability) of the plasma is investigated, first at arbitrary coupling constant Γ via an equivalent 2D Euclidean-field theory, and then at the specific values of Γ = 2*integer via an equivalent 1D fermionic model. The answer to the question in the title is that there is strong evidence for the model being not exactly solvable at arbitrary Γ but becoming exactly solvable at Γ = 2*integer. As a by-product of the developed formalism, the gauge invariance of the plasma is proven at the free-fermion point Γ = 2; the related mathematical peculiarity is the exact inversion of a class of infinite-dimensional matrices.