Gaussian integral with respect to Grassmann variables

Question

Let $A$ be an antisymmetric matrix of even dimension $n$ and $\theta$ be a column vector consisting of $n$ Grassmann variables $\theta_i$. Then the solution of the integral $$\int d\theta_1\dots d\theta_n \exp(-\frac{1}{2}\theta^T A\theta)\tag{1}$$ is given by ${\rm Pf} (A)$. My question is: What is the solution of the integral if it is not assumed that $A$ is antisymmetric.

The origin of my question is a derivation in "Geometry, Topology and Physics" by Nakahara. Nakahara says the solution of $$\prod_{k=1}^N\int d\theta_k^*d\theta_k \,e^{-\theta^\dagger B_N \theta}\tag{2}$$ where $$\theta=\begin{pmatrix}\theta_1\\\vdots\\\theta_N\end{pmatrix},\,\theta^\dagger=\begin{pmatrix}\theta_1^*,\dots,\theta_N^*\end{pmatrix},\,B_N=\begin{pmatrix} 1 & 0 & \dots & 0 & -y \\ y & 1 & 0 &\dots 0 & \\ 0 & y & 1 & \dots 0& \\ \vdots & & \ddots & \ddots & \vdots \\0&0&\dots&y&1 \end{pmatrix}\tag{3}$$ is given by $\det B_N$. However $B_N$ is not antisymmetric. How does Nakahara solve this integral?

Answer 1

1. On one hand, the argument of the exponential in eq. (1) effectively only sees the antisymmetric part of $A$, so one just has to replace $A$ with its antisymmetric part.

2. On the other hand in the complex case (2), one can effectively treat $\theta$ and $\theta^{*}$ as independent variables, so here there is no antisymmetric requirement for the $B_N$ matrix.