The Ratio of Two Zeta-determinants of Dirac Laplacians Associated with Unitary Involutions on a Compact Manifold with Cylindrical End
نویسنده
چکیده
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 Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler’s gluing formula for zeta-determinants and the scattering theory developed by W. Müller in [M]. This result was also obtained independently by J. Park and K. Wojciechowski ([PW2]).
منابع مشابه
On 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 کاملDecomposition of the Ζ-determinant for the Laplacian on Manifolds with Cylindrical End
In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula expl...
متن کاملInverse Problem for Interior Spectral Data of the Dirac Operator with Discontinuous Conditions
In this paper, we study the inverse problem for Dirac differential operators with discontinuity conditions in a compact interval. It is shown that the potential functions can be uniquely determined by the value of the potential on some interval and parts of two sets of eigenvalues. Also, it is shown that the potential function can be uniquely determined by a part of a set of values of eigenfun...
متن کاملAdiabatic Decomposition of the Ζ-determinant and Scattering Theory
We discuss the decomposition of the ζ-determinant of the square of the Dirac operator into contributions coming from different parts of the manifold. The “easy” case was worked out in paper [27]. Due to the assumptions made on the operators in [27], we were able to avoid the presence of the “small eigenvalues” which provide the large time contribution to the determinant. In the present work we ...
متن کاملDirac submanifolds and Poisson involutions
Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds i...
متن کامل