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

The model under consideration is the two-dimensional (2D) one-component plasma of pointlike 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 \Gamma via an equivalent 2D Euclidean-field theory, and then at the specific values of \Gamma=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 \Gamma but becoming exactly solvable at \Gamma=2*integer. As a by-product of the developed formalism, the gauge invariance of the plasma is proven at the free-fermion point \Gamma=2; the related mathematical peculiarity is the exact inversion of a class of infinite-dimensional matrices.