Abstract: We consider a flat complex vector bundle over a compact Riemannian manifold with smooth boundary. We suppose it admits a fiber wise non-degenerate symmetric complex bilinear form. We define non self-adjoint Laplacians acting on (vector valued) differential forms and impose absolute/relative boundary conditions. This is an elliptic boundary value problem. We are interested in the variation of the corresponding complex valued Ray--Singer torsion with respect to a variation of the bilinear form. It turns out that this amounts to compute the constant term in the asymptotic expansion of the corresponding heat kernel. In the talk, we present a method to compute such a term by noting that this is a locally computable invariant and using results from the Hermitian setting.