Application of "real" Grassmann Gaussian integrals

Question

In Appendix 2B of the CFT yellow book by Francesco et al, the authors introduced two types of Grassmann Gaussian integrals (the $\theta$'s below are generators of a Grassmann algebra):

• The "real" one

$$ I = \int d\theta_1 \cdots d\theta_n \exp(-\frac{1}{2} \theta^T A \theta) \tag{2.223} $$

• The "complex" one

$$ I_2 = \int d\bar{\theta} d\theta \, \exp(-\bar{\theta} M \theta) \tag{2.231} $$

$$ d\bar{\theta} d\theta = \prod_{i=1}^n d\bar{\theta}_i d\theta_i \tag{2.232} $$

I know that the "complex" is useful because its generalization to functional integral is the coherent state path integral for free fermions. But I have not encountered applications of the "real" integral, and did not search the yellow book for it. Can anyone kindly tell me applications of the "real" Grassmann Gaussian integral? Is it related to Majorana fermions (which is "real") or some other coherent state path integral?

Answer 1

The "real" integral evalautes to the Pfaffian of $A$ where for an $2n$-by$2n$ skew symmetric matrix $A$ $$ {\rm Pf}A= \frac 1{2^n n!} \epsilon_{i_1, \ldots, i_{2n}} A_{i_1,i_2}\ldots A_{2n-1,2n}. $$
The Pfaffian has the property that $({\rm Pf} A)^2= {\rm det}A$, and has applications many places in combinatorics. For Majorana fermion path integrals it replaces the one-loop Matthews-Salam determinant.

2:

Answer 2

1. Real Grassmann variables are used all over physics.

   Examples:

   • Majorana spinors,

   • Faddeev-Popov ghosts $c,\bar{c}$. (The bar denotes negative ghost number; not complex conjugation.)

2. Gaussian integration of real Grassmann variables are useful in defining the Pfaffian, cf. e.g. my Math.SE post here. In turn the Pfaffian appears in many areas of physics.

   Examples:

   • Various measure factors of the Hamiltonian phase space path integral.

   • The Cachazo-He-Yuan (CHY) representation.