Burghelea-Friedlander-Kappeler's gluing formula and the adiabatic decomposition of the zeta-determinant of a Dirac Laplacian

Abstract

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 Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions. 1991 Mathematics Subject Classification. 58J52, 58J50.

The zeta-determinants of Dirac Laplacians with boundary conditions on the smooth, self-adjoint Grassmannian

“…Then it is not difficult to show (cf. [13] or [14]) that ker (Q 1 + |B|) = φ| Y | φ ∈ L 2,M 1,∞ + L ext 2,M 1,∞ = Im C 1 ∩ Im Π >,C(0) + .…”

“…Then (−∂ 2 u + B 2 ) N 0,r ,γ 0 ,γ r and (−∂ 2 u + B 2 ) N 0,r ,γ 0 ,Π <,σ − are invertible operators and Q 1 is expressed as (cf. [14])…”

“…Using the fact that Q − |B| [13] and P − Π > are smoothing operators, we can show that the zeta-determinant of…”

Gluing Formulae of Spectral Invariants and Cauchy Data Spaces

“…where u denotes the variable of the normal direction to Y and ∆ Y is a Laplace type operator over Y , we can obtain the exact value of C(Y ) as in [9], [15], [28],…”

Combinatorial Quantum Field Theory and Gluing Formula for Determinants

“…Investigation of the cut and paste behaviour of zeta-regularized determinants has been initiated by Forman [For92] and Burghelea-Friedlander-Kappeler in [BFK92]. Their studies have triggered further analysis by various authors including Park-Wojciechowski [PaWo05] and Lee [Lee03].…”