Interacting SPT phases are not Morita invariant

Abstract

Class D topological superconductors have been described as invertible topological phases protected by charge Q and particle-hole symmetry C. A competing description is that class D has no internal symmetries except for the fermion parity group Z F 2 = {1, (−1) F }. In the weakly interacting setting, it can be argued that 'particle-hole symmetry cancels charge' in a suitable sense. Namely, the classification results are independent of which of the two internal symmetry groups are taken because of a Morita equivalence. However, we argue that for strongly interacting particles, the group of symmetry-protected topological phases in the two cases are nonisomorphic in dimension 2 + 1. This shows that in contrast to the free case, interacting phases are not Morita invariant. To accomplish this, we use the approach to interacting phases using invertible field theories and bordism. We give explicit expressions of invertible field theories which have the two different groups Z F 2 and U (1)Q ⋊ Z C 2 as internal symmetries and give spacetime manifolds on which their partition functions disagree. Techniques from algebraic topology are used to compute the relevant bordism groups, most importantly the James spectral sequence. The result is that there are both a new Z2-and a new Z-invariant for U (1)Q ⋊ Z F 2 that are not present for Z F 2 .