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 an assumption that the product structure is given near the 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.

We discuss the decomposition of the ζ-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold. The result was announced in [16] . The proof sketched in [16] was based on results of Brüning and Lesch (see [4]). In the meantime we have found another proof, more direct and elementary, and closer to the spirit of the original papers which initiated the study of the adiabatic decomposition of the spectral invariants (see [7] and [21]). We discuss this proof in detail. We study the general case (non-invertible tangential operator) in forthcoming work (see [17] and [18]). In the Appendix we present the computation of the cylinder contribution to the ζ-function of the Dirac Laplacian on a manifold with boundary, which we need in the main body of the paper. This computation is also used to show the vanishing result for the ζ-function on a manifold with boundary. ResultsLet D : C ∞ (M; S) → C ∞ (M; S) be a compatible Dirac operator acting on sections of a bundle of Clifford modules S over a closed manifold M. Assume that we have a decomposition of M as M 1 ∪ M 2 , where M 1 and M 2 are compact manifolds with boundary such thatThe ζ-determinant of the operator D is given by the formula (0.2) det ζ D = e iπ 2 (ζ D 2 (0)−η D (0)) ·e

In this paper we describe the difference of log of two zeta-determinants of Dirac Laplacians subject to the Dirichlet boundary condition and a boundary condition on the smooth, self-adjoint Grassmannian Gr * ∞ (D) on a compact manifold with boundary. Using this result we obtain the result of Scott and Wojciechowski [S.G. Scott, Zeta determinants on manifolds with boundary, J. Funct. Anal. 192 (2002) 112-185; S.G. Scott, K.P. Wojciechowski, The ζ -determinant and Quillen determinant for a Dirac operator on a manifold with boundary, Geom. Funct. Anal. 10 (2000) 1202-1236] concerning the quotient of two zeta-determinants of Dirac Laplacians with boundary conditions on Gr * ∞ (D). We apply these results to the BFK-gluing formula to obtain the gluing formula for the zeta-determinants of Dirac Laplacians with respect to boundary conditions on Gr * ∞ (D). We next discuss the zetadeterminants of Dirac Laplacians subject to the Dirichlet or APS boundary condition on a finite cylinder and finally discuss the relative zeta-determinant on a manifold with cylindrical end when the APS boundary condition is imposed on the bottom of the cylinder.

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The Burghelea-Friedlander-Kappeler (BFK)-gluing formula for the regularized zeta-determinants of Laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta-determinants of a one-parameter family of the Dirichlet-to-Neumann operators. When the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface, this constant is well known. In this paper, we discuss this constant in two cases: one is when a warped product metric is given near a cutting hypersurface, and the other is when a manifold is a product manifold. Especially in the first case, we use the result of Fucci and Kirsten [Commun. Math. Phys. 317, 635-665 (2013)] in which the regularized zeta-determinant of the Laplacian defined on a warped product manifold is computed.