Fermionic path integral for exact enumeration of polygons on the simple cubic lattice

Enumerating polygons on regular lattices is a classic problem in rigorous statistical mechanics. The goal of enumerating polygons on the square lattice via fermionic path integration was achieved using a free-fermion quadratic action in the late 1970s. Given that polygon edges only link two vertices, it is considered plausible, if not natural, that an action of degree 2 in the Grassmann variables might suffice to enumerate lattice polygons in any dimension. Nevertheless, on nonplanar lattices the problem has remained open for more than four decades. Here, we derive the Grassmann action for exact enumeration of polygons on the simple cubic lattice. Moreover, we prove that this action is not quadratic but quartic, corresponding to a model of interacting fermions.