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Dual Fermion Condensates in Curved Space

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.