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

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

2010-04-11

arxiv.org/abs/1004.1753v1

Mathematics

Differential Geometry

39 pages

Scientific paper

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary. For this purpose we introduce two boundary conditions which are complementary to each other and fit to the odd signature operator $\mathcal{B}$. We show that they are well-posed boundary conditions for $\mathcal{B}$ in the sense of Seeley and show that the zeta-determinants for $\mathcal{B}^{2}$ and eta-invariants for $\mathcal{B}$ 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.

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