Chern-Simons theory, exactly solvable models and free fermions at finite temperature

TL;DR

This paper reveals that Chern-Simons matrix models can be viewed as exactly solvable 1D models, connecting them to free fermions at finite temperature and describing their partition functions through Coulomb-like interactions.

Contribution

It establishes a novel interpretation of Chern-Simons matrix models as 1D solvable systems and links them to free fermions and Coulomb plasmas at finite temperature.

Findings

Chern-Simons models are equivalent to 1D exactly solvable models.

Partition functions relate to free fermion density matrices.

Finite temperature effects modeled by Coulomb-like two-body interactions.

Abstract

We show that matrix models in Chern-Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian matrix model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wavefunctions 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 $S^3$ 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.