# A specific integration with Grassmann variables

I have recently read (for example, here) that this relation below is true

$$\int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}),$$ where $$Pf(\mathbf{A})$$ is the Pfaffian of an even dimensional skew-(or anti-)symmetric matrix $$\mathbf{A}$$ and $$\{ z_i \}$$ are Grassmann variables.

Could anyone point out if there is a similar formula for the form below:

$$\int dz \: \big( \frac{1}{2} \sum_{ij} z_i A_{ij} z_j \big)^k,$$ where $$k > 0$$ is an integer?

My guess is yes since

$$e^{\mathbf{A}} = \sum_{j=0}^\infty \frac{\mathbf{A}^j}{j!},$$ but I do not have enough experience with these types of integrals.

Let's denote the number of Grassmann variables $$z_i$$ by $$N$$, i.e., $$i=1,\ldots ,N$$. Then, in your first equation, the result on the right-hand side is exactly generated by the term $$\int dz \: \frac{1}{(N/2)!}\big( \frac{1}{2} \sum_{ij} z_i A_{ij} z_j \big)^{(N/2)}$$ from the exponential series on the left-hand side, since only the monomial containing each $$z_i$$ precisely once yields a non-zero result upon integration. Therefore the result for your desired integral is zero for $$k\neq N/2$$, and $$(N/2)!\ Pf(\mathbf{A})$$ for $$k=N/2$$.