= Fermionic coherent-state trace
{title2=$\operatorname{Tr}A=\int d[\bar\psi,\psi]e^{-\bar\psi\psi}\langle-\psi|A|\psi\rangle$}
The ordinary trace of an even fermionic operator requires $-\psi$ in the bra of its <fermionic coherent state> kernel. In one mode the diagonal matrix elements give $\langle-\psi|A|\psi\rangle=A_{00}-\bar\psi\psi A_{11}$, whose Gaussian <Berezin integral> is $A_{00}+A_{11}$. An untwisted bra gives $A_{00}-A_{11}$, the <supertrace>. This distinction produces the antiperiodic <coherent-state thermal boundary conditions>.
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