  I present here a new method that allows the introduction of a discrete
auxiliary symmetry in a theory in such a way that the eigenvalue spectrum of
the fermion functional determinant is made up of complex conjugated pairs. The
method implies a particular way of introducing and integrating over auxiliary
fields related to a set of artificial shift symmetries. Gauge-fixing the
artificial continuous shift symmetries in the direct and dual sectors leads to
the implementation of direct and dual BRST-type global symmetries and of a
symplectic structure over the field space (as prescribed by the
Batalin-Vilkovisky method). A procedure similar to Kahler polarization in
geometric quantization guarantees the possibility to choose a Kahler structure
over the field space. This structure is generated by a special way of
performing gauge fixing over the direct and dual sectors. The desired discrete
symmetry appears to be induced by the Hodge-* operator. The particular
extension of the field space presented here makes the operators of the de-Rham
cohomology manifest. These become symmetries in the extended theory. This
method implies the identification of the (anti)-BRST and dual-(anti)-BRST
operators with the exterior derivative and its dual in the context of the
complex de-Rham cohomology.
