Parity anomaly
In theoretical physics a quantum field theory is said to have a parity anomaly if its classical action is invariant under a change of parity of the universe, but the quantum theory is not invariant. This kind of anomaly can occur in odd-dimensional gauge theories with fermions whose gauge groups have odd dual Coxeter numbers. They were first introduced by Antti J. Niemi and Gordon Walter Semenoff in the letter Axial-Anomaly-Induced Fermion Fractionization and Effective Gauge-Theory Actions in Odd-Dimensional Space-Times and by A. Norman Redlich in the letter Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions and the article Parity violation and gauge noninvariance of the effective gauge field action in three dimensions. It is in some sense an odd-dimensional version of Edward Witten's SU(2) anomaly in 4-dimensions, and in fact Redlich writes that his demonstration follows Witten's. The anomaly in 3-dimensions Consider a classically parity-invariant gauge theory whose gauge group G has dual Coxeter number h in 3-dimensions. Include n Majorana fermions which transform under a real representation of G. This theory naively suffers from an ultraviolet divergence. If one includes a gauge-invariant regulator then the quantum parity invariance of the theory will be broken if h and n