  We develop a new method that allows us to map models of interacting fermions
onto bosonic models describing collective excitations in an arbitrary
dimension. This mapping becomes exact in the thermodynamic continuous time
limit. The boson models can be written either in the form of a model of
non-interacting bosons in a fluctuating auxiliary field or in the form of a
superfield theory of interacting bosons. We show how one can study the latter
version using perturbation theory. Using the developed diagrammatic technique
we compared the first two orders of perturbation theory with the corresponding
results for the original fermion model and found a perfect agreement. As
concerns the former representation, we suggest a scheme that may be suitable
for Monte Carlo simulations and demonstrate that it is free of the fermionic
sign problem. We discuss in details the properties of the bosonic
representation and argue that there should not be any obstacles preventing from
an efficient computation.
