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In this paper, we discuss the topological classification of time-reversalinvariant topological superconductors. Based on the axion field theorydeveloped in a previous work (Phys. Rev. B ${\bf 87}$ 134519 (2013)), we showhow a simple quantum anomaly in vortex-crossing process predicts a$\mathbb{Z}_{16}$ classification of interacting topological superconductors, inconsistency with other approaches. We also provide a general definition of thequantum anomaly and a general geometric argument that explains the$\mathbb{Z}_{16}$ on more general grounds. Furthermore, we generalize ourapproach to all $4n$ dimensions (with $n$ an integer), and compare our resultswith other approaches to the topological classification.

Axion field theory approach and the classification of interacting topological superconductors