  Let $E$ be a flat complex vector bundle over a closed oriented odd
dimensional manifold $M$ endowed with a flat connection $\nabla$. The refined
analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler.
Recently Mathai and Wu defined and studied the analytic torsion for the twisted
de Rham complex with an odd degree closed differential form $H$, other than one
form, as a flux and with coefficients in $E$. In this paper we generalize the
construction of the refined analytic torsion to the twisted de Rham complex. We
show that the refined analytic torsion of the twisted de Rham complex is
independent of the choice of the Riemannian metric on $M$ and the Hermitian
metric on $E$. We also show that the twisted refined analytic torsion is
invariant (under a natural identification) if $H$ is deformed within its
cohomology class. We prove a duality theorem, establishing a relationship
between the twisted refined analytic torsion corresponding to a flat connection
and its dual. We also define the twisted analogue of the Ray-Singer metric and
calculate the twisted Ray-Singer metric of the twisted refined analytic
torsion. In particular we show that in case that the Hermtitian connection is
flat, the twisted refined analytic torsion is an element with the twisted
Ray-Singer norm one.
