  Recently, Cappell and Miller extended the classical construction of the
analytic torsion for de Rham complexes to coupling with an arbitrary flat
bundle and the holomorphic torsion for $\bar{\partial}$-complexes to coupling
with an arbitrary holomorphic bundle with compatible connection of type
$(1,1)$. Cappell and Miller studied the properties of these torsions, including
the behavior under metric deformations. On the other hand, Mathai and Wu
generalized the classical construction of the analytic torsion to the twisted
de Rham complexes with an odd degree closed form as a flux and later, more
generally, to the $\mathbb{Z}_2$-graded elliptic complexes. Mathai and Wu also
studied the properties of analytic torsions for the $\mathbb{Z}_2$-graded
elliptic complexes, including the behavior under metric and flux deformations.
In this paper we define the Cappell-Miller holomorphic torsion for the twisted
Dolbeault-type complexes and the Cappell-Miller analytic torsion for the
twisted de Rham complexes. We obtain variation formulas for the twisted
Cappell-Miller holomorphic and analytic torsions under metric and flux
deformations.
