Topological insulators avoid the parity anomaly

The surface of a 3topological insulator hosts an odd number of gapless Dirac fermions when charge conjugation and time-reversal symmetries are preserved. Viewed as a purely 2 system, this surface theory would necessarily explicitly break parity and time-reversal when coupled to a fluctuating gauge field. Here, we explain why such a state can exist on the boundary of a 3 system without breaking these symmetries, even if the number of boundary components is odd. This is accomplished from two complementary perspectives: topological quantization conditions and regularization. We first discuss the conditions under which (continuous) large gauge transformations may exist when the theory lives on a boundary of a higher-dimensional space-time. Next, we show how the higher-dimensional bulk theory is essential in providing a parity-invariant regularization of the theory living on the lower-dimensional boundary or defect.