  We construct a canonical element, called the refined analytic torsion, of the
determinant line of the cohomology of a closed oriented odd-dimensional
manifold M with coefficients in a flat complex vector bundle E. We compute the
Ray-Singer norm of the refined analytic torsion. In particular, if there exists
a flat Hermitian metric on E, we show that this norm is equal to 1. We prove a
duality theorem, establishing a relationship between the refined analytic
torsions corresponding to a flat connection and its dual.
