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Twisted Cappell-Miller holomorphic and analytic torsions

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.