paritytwisted

Paritytwisted is a term used in mathematics and theoretical physics to describe a modification of the standard parity operation by a twisting mechanism that alters how parity interacts with other symmetries. It is used to describe systems where a parity transformation does not simply commute with, or may acquire a phase when composed with, another symmetry operation. The concept is often discussed in the context of representations of symmetry groups, topological phases, and conformal field theories.

In formal terms, paritytwisted constructions introduce a twist, typically encoded by a 2-cocycle or a parity-dependent sign, such that the action of a symmetry group G on a vector space V is projective rather than linear. If ε: G -> {±1} denotes a parity homomorphism, a paritytwisted representation satisfies ρ(g)ρ(h) = α(g,h) ρ(gh) with a twist factor α determined by parity information. When α takes the value -1 on certain parity-conserving pairs, the representation can capture statistics or anomaly-like behavior. In many treatments, the twist is constrained so that ε appears in the compatibility condition with ρ, ensuring associativity of the action.

Applications of paritytwisted constructions include the study of symmetry-enriched topological phases, orbifold and twisted sectors in conformal field theories, and the representation theory of supergroups where fermionic parity alters the sign of certain maps. Simple examples include the cyclic group of order two, where the generator representing parity acts with a square that may be ± identity depending on the chosen twist.

See also parity, projective representation, twisted group algebra, and symmetry fractionalization.