Interacting topological phases and modular invariance

TL;DR

This paper explores the relationship between interacting topological phases of (2+1)D superconductors, modular invariance, and gravitational anomalies, revealing a $ ext{Z}_8$ classification linked to edge state stability.

Contribution

It demonstrates how interactions affect topological classifications and connects these effects to modular invariance and gravitational anomalies in superstring theory.

Findings

Edge states with $N_f=8$ become unstable due to interactions.

The $ ext{Z}_8$ classification arises from gravitational anomaly considerations.

Modular invariance constrains the stability and classification of topological phases.

Abstract

We discuss a (2+1) dimensional topological superconductor with $N_f$ left- and right-moving Majorana edge modes and a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry. In the absence of interactions, these phases are distinguished by an integral topological invariant $N_f$. With interactions, the edge state in the case $N_f=8$ is unstable against interactions, and a $\mathbb{Z}_2\times \mathbb{Z}_2$ invariant mass gap can be generated dynamically. We show that this phenomenon is closely related to the modular invariance of type II superstring theory. More generally, we show that the global gravitational anomaly of the non-chiral Majorana edge states is the physical manifestation of the bulk topological superconductors classified by $\mathbb{Z}_8$.