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

• Published in: Manuscripta mathematica Vol. 139; no. 1-2; pp. 91 - 122
• Main Authors: Huang, Rung-Tzung, Lee, Yoonweon
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.09.2012
• Subjects: Algebraic Geometry; Calculus of Variations and Optimal Control; Optimization; Geometry; Lie Groups; Mathematics; Mathematics and Statistics; Number Theory; Topological Groups; Primary: 58J52; Secondary: 58J28; 58J50
• ISSN: 0025-2611; 1432-1785
• DOI: 10.1007/s00229-011-0504-3

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions and 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 and . 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 or 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.