Real Grassmann Gaussian integral ($\int d^{2m}\omega e^{-{\omega }^{T}A\omega /2}\propto \mathrm{Pf}(A)$)

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 $\mathrm{Pf}(A{)}^2=\det A$ determines its square but does not discard its orientation.

For an even-dimensional antisymmetric matrix $M$, put $\theta =(u+iv)/\sqrt{2}$ and $\bar{\theta }=(u-iv)/\sqrt{2}$, with independent Grassmann variables $u,v$. Anticommutation cancels the mixed terms, giving ${\bar{\theta }}^{T}M\theta =(u^{T}Mu+v^{T}Mv)/2$. The constant Grassmann change-of-variables formula changes only the normalization. The two real Pfaffians multiply to $\mathrm{Pf}(M{)}^2=\det M$, agreeing with the complex Grassmann Gaussian integral. In odd dimension $\det M=0$ and both top-degree integrals vanish.