Parity shift - Knowlpedia

Core idea

For a super vector space $V=V_{\bar0}\oplus V_{\bar1}$ , its parity shift or parity reversal $\Pi V$ is defined by

(\Pi V)_{\bar0}=V_{\bar1}, \qquad (\Pi V)_{\bar1}=V_{\bar0}.

On even linear maps, $\Pi f:\Pi V\to\Pi W$ has the same underlying linear map as $f$ . This defines an even functor $\Pi$ with $\Pi^2\cong\operatorname{id}$ .

Odd maps as even maps

An odd linear map $f:V\to W$ can be regarded as an even map

V\longrightarrow\Pi W \qquad\text{or}\qquad \Pi V\longrightarrow W.

This is one reason parity shift is useful: it converts degree-one data into ordinary degree-zero morphisms. Formulas involving shifted tensor products may place signs in the identifications above; authors should state the chosen sign convention when those identifications enter a calculation.

No canonical even identification

The underlying ungraded vector spaces of $V$ and $\Pi V$ are the same, but the identity of that underlying vector space is odd, not even. Consequently there is generally no canonical isomorphism $V\cong\Pi V$ in the category of super vector spaces.

References

V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, American Mathematical Society, 2004. DOI record. Relevant: Section 1.1.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, European Mathematical Society, 2011. DOI record. Relevant: Chapter 1.