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The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the deformation method used in [6], when the odd signature operator comes from a Hermitian flat connection and all de Rham cohomologies vanish.