  The surface of a 3+1d topological insulator hosts an odd number of gapless
Dirac fermions when charge conjugation and time-reversal symmetries are
preserved. Viewed as a purely 2+1d 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+1d 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 spacetime.
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.
