Fermion number operator ($F$)

The fermion number operator counts occupied fermionic modes. Its parity $(-1{)}^F$ is positive on even exterior degree and negative on odd exterior degree.

A supertrace is a trace weighted by the grading of a Hilbert space. For fermion parity $(-1{)}^F$, it is $\mathrm{Str}A=\mathrm{Tr}[(-1{)}^{F}A]$: even states contribute with a plus sign and odd states with a minus sign.

In supersymmetric quantum mechanics, paired positive-energy states have opposite fermion parity and cancel in the supertrace of $e^{-\beta H}$. Unpaired zero modes then contribute to the Witten index. This cancellation requires a well-defined trace or a consistent regulator; a continuum reaching zero energy can invalidate a naive term-by-term argument. The supertrace is therefore useful for counting protected differences between sectors, while an ordinary thermal trace adds both sectors and counts a different quantity.

An equivariant supertrace inserts a symmetry $f$ as well as fermion parity, $\mathrm{Tr}((-1{)}^{F}fA)$. In a path integral, $f$ produces boundary conditions twisted by its action.