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

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