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 ∂-complexes to coupling with an arbitrary holomorphic bundle with
compatible connection of type (1,1). Cappell and Miller also studied the behavior
of these torsions 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 ℤ2-graded elliptic complexes. Mathai and Wu
also studied the properties of analytic torsions for the ℤ2-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.
