= Supertrace
{title2=$\operatorname{Str}A=\operatorname{Tr}((-1)^FA)$}
A supertrace is a trace weighted by the grading of a <Hilbert space>. For <fermion parity> $(-1)^F$, it is $\operatorname{Str}A=\operatorname{Tr}[(-1)^F A]$: 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.
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