Bose-Fermi Equivalence in Three Dimensional Non-commutative Space-Time

TL;DR

This paper explores the fermionization of a non-commutative scalar field theory coupled with a gauge field, revealing a non-local dual fermionic theory that reduces to the commutative case as non-commutativity vanishes.

Contribution

It provides a non-perturbative derivation of the dual fermionic theory for a non-commutative scalar-gauge system using the Polyakov spin factor formalism.

Findings

Dual fermionic theory is non-local

Non-commutative effects persist in the dual theory

Commutative limit recovers known results

Abstract

We study the Fermionisation of Seiberg-Witten mapped action (to order $\theta$) of the $\lambda\phi^{4}$ theory coupled minimally with U(1) gauge field governed by Chern-Simon action. Starting from the corresponding partition function we derive non-perturbatively (in coupling constant) the partition function of the spin 1/2 theory following Polyakov spin factor formalism. We find the dual interacting fermionic theory is non local. This feature persist also in the limit of vanishing self coupling. In $\theta\to 0$ limit, the commutative result is regained.