# Gaussian integral with respect to Grassmann variables

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?

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.