Interacting topological phases and modular invariance

We discuss (2+1)-dimensional topological superconductors 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 of $N_f=8$ 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 nonchiral Majorana edge states is the physical manifestation of the bulk topological superconductors classified by ${\mathbb{Z}}_8$.