Burghelea-friedlander-kappeler’s Gluing Formula and the Adiabatic Decomposition of the Zeta-determinant of a Dirac Laplacian
نویسنده
چکیده
In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zetadeterminant given by Burghelea, Friedlander and Kappeler with some assumptions, we prove the adiabatic decomposition theorem of the zeta-determinant of a Dirac Laplacian. This result was originally proved by J. Park and K. Wojciechowski in [11] but our method is completely different from the one they presented. §
منابع مشابه
Burghelea-friedlander-kappeler’s Gluing Formula for the Zeta-determinant and Its Applications to the Adiabatic Decompositions of the Zeta-determinant and the Analytic Torsion
The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under the assumption of the product structure near boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dir...
متن کاملThe Ratio of Two Zeta-determinants of Dirac Laplacians Associated with Unitary Involutions on a Compact Manifold with Cylindrical End
Abstract. Given two unitary involutions σ1 and σ2 satisfying Gσi = −σiG on kerB on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. Müller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with σ1 and σ2. In this paper we establish the analogous formula for the zeta-determinants of Dirac Lap...
متن کاملAdiabatic Decomposition of the Ζ-determinant and Dirichlet to Neumann Operator
Abstract. This paper is companion to our earlier work [8] (see also announcement [7]). Let M be a closed manifold and Y be an embedded hypersurface, such that there is a decomposition of M = M1 ∪M2 into two manifolds with boundary M1 and M2 , with M1 ∩M2 = Y . In [8] we proved the decomposition formula for detζ∆ the ζ-determinant of a Dirac Laplacian ∆ on M . The contributions coming from M1 an...
متن کاملJ an 2 00 3 Determinant bundles , boundaries , and surgery Ulrich Bunke and Jinsung Park February
In this note we specialize and illustrate the ideas developed in the paper [4] Families, Eta forms, and Deligne cohomology in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regularized determinant of the Dirac Laplacian in [11].
متن کاملOn Gluing Formulas for the Spectral Invariants of Dirac Type Operators
In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs. 1. The gluing problem for the spectral invariants Since their inception, the eta invariant and the ζ-determinant of Dirac type operators have influenced mathematics and physics in...
متن کامل