Interacting topological phases and modular invariance

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We discuss a (2+1) dimensional topological superconductor with Nf left- and right-moving Majorana edge modes and a Z₂× Z₂ symmetry. In the absence of interactions, these phases are distinguished by an integral topological invariant Nf. With interactions, the edge state in the case Nf=8 is unstable against interactions, and a Z₂× Z₂ 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 Z₈.