Parityshifted

Parityshifted is a term used in mathematics and theoretical physics to describe objects obtained by applying the parity shift to a graded structure, typically a Z2-graded vector space, module, or algebra. The idea is to swap the two parity components of the object, changing which elements are considered even and which are odd.

Concretely, if V is a Z2-graded vector space V = V0 ⊕ V1, the parityshifted object, often denoted ΠV, has components defined by (ΠV)0 = V1 and (ΠV)1 = V0. The underlying vector space is the same, but the grading is flipped. The parity-shift functor Π is an involutive auto-equivalence, meaning Π(ΠV) is naturally isomorphic to V.

Morphisms respect the shifted parity as well. If f: V → W is a homogeneous linear map of parity p (even if p = 0, odd if p = 1), then the induced map Πf: ΠV → ΠW has parity p + 1 (mod 2). This shift in parity affects sign conventions in graded tensor products and in the composition of maps, where signs depend on the parity of the objects involved.

Parityshifted objects play a central role in the theory of superalgebras, supergeometry, and related categorical constructions. They are used to model odd elements and to relate even and odd components in a uniform framework. The concept also appears in derived and triangulated categories, BRST cohomology, and mathematical formulations of supersymmetry, where the parity shift clarifies the behavior of morphisms and tensor structures under parity changes.