Interacting topological phases and modular invariance

TL;DR

The paper examines a (2+1)D topological superconductor with $N_f$ left- and right-moving Majorana edge modes under a $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry, showing that while non-interacting phases are labeled by $N_f$, interactions render the $N_f=8$ edge unstable and reveal a bulk $\mathbb{Z}_8$ classification. A central methodology is the use of modular invariance and global gravitational anomalies on the edge to diagnose bulk topology, including a detailed analysis of large gauge and large diffeomorphism transformations and symmetry projections. The key contributions include an explicit $\mathrm{SO}(8)$-based interacting edge term that gaps the edge without breaking symmetry, a demonstration that modular invariance requires specific projections analogous to GSO in type II string theories, and the identification of a deep connection between edge modular properties and bulk $\mathbb{Z}_8$ topology. The findings provide a robust framework for understanding stability of non-chiral topological phases under interactions and highlight the role of modular invariance as a diagnostic tool with potential implications for broader symmetry classes and lattice realizations.

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 $\mathbb{Z}_8$.