  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 2 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.
