Interacting topological phases and quantum anomalies

We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the 'collapse' of the non-interacting classification, where an integer classification ('Z') at non-interacting level reduces to Z(8) or Z(16) in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.