Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions

Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the non-interacting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)-dimensional fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\times$ reflection) and electromagnetic $\mathrm{U}(1)$ symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large $\mathrm{U}(1)$ gauge transformations, but picks up an anomalous sign, signaling a $\mathbb{Z}_2$ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, time-reversal symmetric topological superconductors, we discuss the invariance/non-invariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (3d modular transformations). The connection to the collapse of the non-interacting classification by an integer ($\mathbb{Z}$) to $\mathbb{Z}_{16}$, in the presence of interactions, is discussed.