Fermionic coherent-state trace ($\mathrm{Tr}A=\int d[\bar{\psi },\psi ]e^{-\bar{\psi }\psi }⟨-\psi |A|\psi ⟩$)

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 $⟨-\psi |A|\psi ⟩=A_{00}-\bar{\psi }\psi A_{11}$, 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.