The refined analytic torsion and a well-posed boundary condition for the odd signature operator

Abstract

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator $\mathcal{B}$ in the sense of Seeley. We then show that the zeta-determinants of ${\mathcal{B}}^2$ and eta-invariants of $\mathcal{B}$ subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined analytic torsion on a compact manifold with boundary and show that it is invariant on the change of metrics in the interior of the manifold. We finally describe the refined analytic torsion under these boundary conditions as an element of the determinant line.

33 pages, the paper is considerably rewritten, main assumptions changed

References in corpus (3)

• Complex valued Ray-Singer torsion
• Complex valued Ray--Singer torsion II
• Gluing Formula for Refined Analytic Torsion

Cited by in corpus (4)

• The gluing formula of the refined analytic torsion for an acyclic Hermitian connection
• The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary
• Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions
• The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions