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 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.

Mobility edge and Black Hole Horizon

“…• The second one-matrix model involves the potential providing the weak confinement [30] of eigenvalues. Such type of model was used in the matrix model description of Chern-Simons theory [37] and was solved in terms of the q-Hermite polynomials. Asymptotically potential behaves as…”

Supersymmetric gauge theories, Coulomb gases, and Chern-Simons matrix models

“…This is true of the N = 4 theories on S 3 that arise as the low-energy limit of N = 4 supersymmetric Yang-Mills theory in four dimensions [27], which is also connected to the six-dimensional (2, 0) superconformal theories via certain dimensional reductions. It is also the case of Chern-Simons gauge theory [15]. For the one-dimensional wavefunction…”

“…the same direct approach in [15] can be used to find the general Hamiltonian of a bosonic model for which (2.45) is a ground state. It can also be found as a limit of the Hamiltonian for the Chern-Simons fermionic model (1.4) which in its most general version is characterized by a ground state wavefunction [15] (2.46) Ψ (m)…”

“…The computation of (2.1) is also intimately related to an old statistical mechanics problem studied in detail by Baxter in 1963 [18]. While the matrix model of U (N ) Chern-Simons gauge theory on S 3 has an interpretation as a one-component Coulomb plasma living on the surface of a cylinder in Dyson's Coulomb gas picture of random matrix ensembles [15], the expression (2.1) is the partition function of a one-dimensional Coulomb gas known as a one-dimensional jellium model [18]: The two-body interaction |x i − x j | is a Coulomb potential in one dimension, while log sinh(|x i − x j |/2R c ) is the Coulomb potential on a cylinder of radius R c . The strong coupling limit that maps (1.3) to (2.1) is then a thin cylinder limit R c → 0 in the Coulomb plasma representation.…”

“…A key interpretation that we advocate from this relationship between the two apparently distinct gauge theories is through their natural appearences in the theory of one-dimensional exactly solvable models. The partition function (1.3) of Chern-Simons theory can be interpreted as the L 2 -norm of the ground state wavefunction of a fermionic model on a cylinder of radius R c = 1 with Hamiltonian [15] (1.4)…”