Parity Anomaly and Duality Web

TL;DR

This paper reviews the parity anomaly in 2+1 and 1+1 dimensions and explores the duality web in 2+1 dimensions, including its extension to spin$_c$ manifolds and connections to electric-magnetic duality.

Contribution

It provides a comprehensive review of the parity anomaly and extends the duality web to spin$_c$ manifolds, addressing global effects and consistency issues.

Findings

Demonstrates the parity anomaly in 3D and 1D Dirac fermion theories.

Extends the duality web to spin$_c$ manifolds to resolve spin structure inconsistencies.

Connects the 2+1 dimensional duality web with electric-magnetic duality in 3+1 dimensions.

Abstract

We review the parity anomaly and a duality web in 2+1 dimensions. An odd dimensional non-interacting Dirac fermion theory is not parity invariant at quantum level. We demonstrate the parity anomaly in a three dimensional non-interacting Dirac fermion theory and a one dimensional non-interacting Dirac fermion theory. These theories can generate non-gauge invariant Abelian Chern-Simons terms at a finite temperature through an effective action. The parity anomaly also leads us to study the duality web in 2+1 dimensions, in which the 2+1 dimensional duality web begins from the conjecture of a duality between a three dimensional Dirac fermion theory and a three dimensional interacting scalar field theory at the Wilson-Fisher fixed point. We first review the duality web for the flat background, then we discuss its extension to the spin$_c$ manifold to avoid inconsistency from the spin structure. This also leads a global effect. We discuss that the composite fermions approach of the quantum Hall system also suffers from the same issue of the global description. Finally, we use perspective of the electric-magnetic duality of the Abelian gauge theory in 3+1 dimensions to study the 2+1 dimensional duality web.