We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the ddd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar ϕ4ϕ4\phi^4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field ϕ(x)∈Cϕ(x)∈C\phi(x)\in \mathbb C (standard formulation) or a nilpotent one satisfying ϕ(x)2=0ϕ(x)2=0\phi(x)^2 =0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.