= Real Grassmann Gaussian integral
{title2=$\int d^{2m}\omega\,e^{-\omega^TA\omega/2}\propto\operatorname{Pf}(A)$}
For an <antisymmetric matrix> $A$, a <Berezin integral> extracts the top coefficient of the exponential. Expansion into pairings gives the <Pfaffian>, with a constant sign depending on the order of the differentials. This is a polynomial in the entries of $A$, not the positive square root of its <determinant>: in dimension two it is proportional to $A_{12}$ and changes sign with that entry. The identity $\operatorname{Pf}(A)^2=\det A$ determines its square but does not discard its orientation.
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