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

TL;DR

This paper establishes a gluing formula for the refined analytic torsion on manifolds with boundary, connecting it to known invariants like Ray-Singer torsion and eta invariant under specific boundary conditions.

Contribution

It extends the gluing formula to the refined analytic torsion for acyclic Hermitian connections with new boundary conditions, linking it to classical invariants.

Findings

Derived the gluing formula for refined analytic torsion with specific boundary conditions.

Compared refined torsion with classical invariants under different boundary conditions.

Connected the refined torsion to Ray-Singer torsion and eta invariant through boundary condition analysis.

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.