A Duality Web in 2+1 Dimensions and Condensed Matter Physics

TL;DR

This paper develops a web of dualities in 2+1 dimensions connecting fermionic and bosonic theories, with implications for condensed matter phenomena like topological insulators and quantum Hall effects.

Contribution

It introduces a generalized duality web in 2+1 dimensions, clarifies the realization of time reversal, and links these dualities to higher-dimensional S-duality, advancing understanding in both high energy and condensed matter physics.

Findings

Identifies dualities between fermionic and bosonic theories in 2+1D.

Resolves issues regarding time reversal in these theories.

Connects dualities to 3+1D S-duality and applications in condensed matter physics.

Abstract

Building on earlier work in the high energy and condensed matter communities, we present a web of dualities in $2+1$ dimensions that generalize the known particle/vortex duality. Some of the dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. For example, the long distance behavior of the $2+1$-dimensional analog of QED with a single Dirac fermion (a theory known as $U(1)_{1/2}$) is identified with the $O(2)$ Wilson-Fisher fixed point. The gauged version of that fixed point with a Chern-Simons coupling at level one is identified as a free Dirac fermion. The latter theory also has a dual version as a fermion interacting with some gauge fields. Assuming some of these dualities, other dualities can be derived. Our analysis resolves a number of confusing issues in the literature including how time reversal is realized in these theories. It also has many applications in condensed matter physics like the theory of topological insulators (and their gapped boundary states) and the problem of electrons in the lowest Landau level at half filling. (Our techniques also clarify some points in the fractional Hall effect and its description using flux attachment.) In addition to presenting several consistency checks, we also present plausible (but not rigorous) derivations of the dualities and relate them to $3+1$-dimensional $S$-duality.