# Matter Chern Simons Theories in a Background Magnetic Field

## TL;DR

This paper exactly solves the fermion propagator in a 2+1D Chern-Simons theory with a magnetic field, revealing shifted Landau levels and confirming duality with a bosonic theory.

## Contribution

It introduces an exact solution to the Schwinger Dyson equation in a non-commutative setting and verifies the Bose-Fermi duality in a magnetic background.

## Key findings

- Exact fermion propagator with shifted Landau levels
- Spectrum matches between fermionic and bosonic dual theories
- Broadening of Landau levels at arbitrary coupling

## Abstract

We study large $N$ 2+1 dimensional fermions in the fundamental representation of an $SU(N)_k$ Chern Simons gauge group in the presence of a uniform background magnetic field for the $U(1)$ global symmetry of this theory. The magnetic field modifies the Schwinger Dyson equation for the propagator in an interesting way; the product between the self energy and the Greens function is replaced by a Moyal star product. Employing a basis of functions previously used in the study of non-commutative solitons, we are able to exactly solve the Schwinger Dyson equation and so determine the fermion propagator. The propagator has a series of poles (and no other singularities) whose locations yield a spectrum of single particle energies at arbitrary t' Hooft coupling and chemical potential. The usual free fermion Landau levels spectrum is shifted and broadened out; we compute the shifts and widths of these levels at arbitrary t'Hooft coupling. As a check on our results we independently solve for the propagators of the conjecturally dual theory of Chern Simons gauged large $N$ fundamental Wilson Fisher bosons also in a background magnetic field but this time only at zero chemical potential. The spectrum of single particle states of the bosonic theory precisely agrees with those of the fermionic theory under Bose-Fermi duality.