CHERN–SIMONS THEORY, EXACTLY SOLVABLE MODELS AND FREE FERMIONS AT FINITE TEMPERATURE

Abstract

We show that matrix models in Chern–Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wave functions and ground-state energies and point out that the models can be interpreted as quasi-1D Coulomb plasmas. We also study the relationship between Chern–Simons theory on S3 and a system of N one-dimensional fermions at finite temperature with harmonic confinement. In particular, we show that the Chern–Simons partition function can be described by the density matrix of the free fermions in a very particular, crystalline, configuration. For this, we both use the Brownian motion and the matrix model description of Chern–Simons theory and find several common features with c = 1 theory at finite temperature. Finally, using the exactly solvable model result, we show that the finite temperature effect can be described with a specific two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at large separations.