  The sign problem in QCD at finite temperature and density leads naturally to
the consideration of complex saddle points of the action or effective action.
The global symmetry $\mathcal{CK}$ of the finite-density action, where
$\mathcal{C}$ is charge conjugation and $\mathcal{K}$ is complex conjugation,
constrains the eigenvalues of the Polyakov loop operator $P$ at a saddle point
in such a way that the action is real at a saddle point, and net color charge
is zero. The values of $Tr_{F}P$ and $Tr_{F}P^{\dagger}$ at the saddle point,
are real but not identical, indicating the different free energy cost
associated with inserting a heavy quark versus an antiquark into the system. At
such complex saddle points, the mass matrix associated with Polyakov loops may
have complex eigenvalues, reflecting oscillatory behavior in color-charge
densities. We illustrate these properties with a simple model which includes
the one-loop contribution of gluons and massless quarks moving in a constant
Polyakov loop background. Confinement-deconfinement effects are modeled
phenomenologically via an added potential term depending on the Polyakov loop
eigenvalues. For sufficiently large $T$ and $\mu$, the results obtained reduce
to those of perturbation theory at the complex saddle point. These results may
be experimentally relevant for the CBM experiment at FAIR.
