Refined analytic torsion as analytic function on the representation variety and applications

We prove that refined analytic torsion on a manifold with boundary is a weakly holomorphic 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 Bruning-Ma gluing formula for the Ray–Singer torsion associated to a non-Hermitian connection. Our proof is quite different from the one given by Bruning and Ma and uses a temporal gauge transformation.