Definition of anticommuting Mean Field Theory and real Grassmann fields

Question

These words come from this article https://arxiv.org/abs/2008.04361, page 6. My question is what is anticommuting Mean Field Theory which appears in the equation (1.1), and what is real Grassmann fields which appears above equation (1.2)? Why the anticommuting Mean Field Theory can be characterized by the equation (1.2)?

Answer 1

You need some info about Grassmann variables and Berezin integration, otherwise the article will be too hard to read. A quick intro is in https://arxiv.org/abs/math/0306396 In particular, you will see there is an analogue of Gaussian measures for these funny anticommuting variables. This is what the authors refer to by anticommuting MFT. Moments of (centered) Gaussians are entirely and explicitly determined by second moments, i.e., the covariance. The same is true here and that is why eq. (1.2) is enough information for the reader.