The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

Abstract: In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}{-, {\mathcal L}{0}}$ and ${\mathcal P}{+, {\mathcal L}{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${\mathcal P}{-, {\mathcal L}{0}}$ and ${\mathcal P}{+, {\mathcal L}{1}}$. In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${\mathcal P}{-, {\mathcal L}{0}}$ or ${\mathcal P}{+, {\mathcal L}{1}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.