Refined analytic torsion as an element of the determinant line

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. 58J52; 58J28, 57R20