  We prove that refined analytic torsion on a manifold with boundary is an
analytic section of the determinant line bundle over the representation
variety. As a fundamental application we establish a gluing formula for refined
analytic torsion on connected components of the complex representation space
which contain a unitary point. Finally we provide a new proof of Bruening-Ma
gluing formula for the Ray-Singer torsion associated to a non-Hermitian
connection. Our proof is quite different from the one given by Bruening and Ma
and uses a temporal gauge transformation.
