  In this paper we compute the effective action at finite temperature and
density for the dual fermion condensate in curved space with the fermions
described by an effective field theory with four-point interactions. The
approach we adopt refines a technique developed earlier to study chiral
symmetry breaking in curved space and it is generalized here to include the
U$(1)$-valued boundary conditions necessary to define the dual condensate. The
method we present is general, includes the coupling between the fermion
condensate and the Polyakov loop, and applies to any ultrastatic background
spacetime with a nonsingular base. It also allows one to include inhomogeneous
and anisotropic phases and therefore it is suitable to study situations where
the geometry is not homogeneous. We first illustrate a procedure, based on heat
kernels, useful to deal with situations where the dual and chiral condensates
(as well as any smooth background field eventually present) are slowly or
rapidly varying functions in space. Then we discuss a different approach based
on the density of states method and on the use of Tauberian theorems to handle
the case of arbitrary chemical potentials. As a trial application, we consider
the case of constant curvature spacetimes and show how to compute numerically
the dual fermion condensate in the case of both homogeneous and inhomogeneous
phases.
