Particle-vortex duality in topological insulators and superconductors

TL;DR

This work analyzes the origins and consequences of duality between topological insulators and topological superconductors in 3+1 and 2+1 dimensions. It shows that in four dimensions the TI–TSC relationship is captured by Maxwell duality via a master action, while in three dimensions the duality arises from self-duality in odd dimensions tied to particle–vortex duality. The authors connect these bulk–boundary dualities to Son's conjecture, deriving a correspondence between a surface Dirac fermion and a composite fermion, including a conductivity relation that follows from the duality. The results provide a unified framework for understanding dual descriptions of topological quantum matter and offer avenues for exploring interacting 3D topological superconductors and related nonperturbative dualities.

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.