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.
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.
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 are odd.
One needs to add n massive Majorana fermions with opposite statistics and take their masses to infinity.
The complication arises from the fact that the 3-dimensional Majorana mass term,
Therefore, the anomalous phase may only be equal to a square root of one, in other words, plus or minus one.
The possibility that it be ill-defined exists because the action contains the fermion kinetic term
The overall phase of the partition function is not an observable in quantum mechanics, and so for a given configuration this sign choice can be made arbitrarily.
, because each time that a pair of eigenvalues changes sign there will be a zero.
Notice that the eigenvalues come in pairs, as discussed for example in Supersymmetric Index Of Three-Dimensional Gauge Theory, and so whenever one eigenvalue crosses zero, two will cross.
These zeroes are counted by the Atiyah–Singer index theorem, which gives the answer h times the second Chern class of the gauge bundle over
If the sign changes an odd number of times then the partition function is ill-defined, and so there is an anomaly.
Using Stokes' theorem and the fact that the exterior derivative of the Chern–Simons action is equal to the instanton number, the 4-dimensional theory on
This implies that the 3-dimensional partition function is ill-defined by a factor of -1 when considering deformations over a path with an odd number of instantons.
The fact that both Chern–Simons terms and Majorana fermions are anomalous under deformations with odd instanton numbers is not a coincidence.
When the Pauli–Villars mass for n Majorana fermions is taken to infinity, Redlich found that the remaining contribution to the partition function is equal to a Chern–Simons term at level −n/2.
This means in particular that integrating out n charged Majorana fermions renormalizes the Chern–Simons level of the corresponding gauge theory by −n/2.