Particle-vortex duality in topological insulators and superconductors

TL;DR

This paper explores the duality between topological insulators and superconductors across different dimensions, revealing connections to particle-vortex duality and proposing a duality involving surface Dirac fermions and composite fermions.

Contribution

It demonstrates how dualities in topological phases can be understood through path integral transformations and relates three-dimensional surface states to four-dimensional bulk properties.

Findings

Duality transformations can be performed at the path integral level in four dimensions.

In three dimensions, the duality manifests as self-duality, related to particle-vortex duality.

A conjecture is supported that surface Dirac fermions are dual to composite fermions.

Abstract

We investigate the origins and implications of the duality between topological insulators and topological superconductors in three and four spacetime dimensions. In the latter, the duality transformation can be made at the level of the path integral in the standard way, while in three dimensions, it takes the form of "self-duality in odd dimensions". In this sense, it is closely related to the particle-vortex duality of planar systems. In particular, we use this to elaborate on Son's conjecture that a three dimensional Dirac fermion that can be thought of as the surface mode of a four dimensional topological insulator is dual to a composite fermion.