We prove a generalization of Kirchhoff’s matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a $q to 0$ limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N-vector model at $N=-1$ or, equivalently, onto the sigma-model taking values in the unit supersphere in $R^{1|2}$. It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.

This is joint work with Sergio Caracciolo, Jesper Jacobsen, Hubert Saleur and Andrea Sportiello.

Department of Physics, New York University