Adiabatic Limits of Eta and Zeta Functions of Elliptic Operators
نویسنده
چکیده
We use the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator δ constructed from an elliptic family of operators with base S. We show that the regularized values η(δt, 0) and tζ(δt, 0) have smooth limits as t → 0, and we identify the limits with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families the functions η(δt, s) and tζ(δt, s) are shown to extend smoothly at t = 0. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms.
منابع مشابه
On the Singularities of the Zeta and Eta Functions of an Elliptic Operator
Let P be a selfadjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s = k m , where k ranges over all non-zero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P w...
متن کاملEta Invariants and Regularized Determinants for Odd Dimensional Hyperbolic Manifolds with Cusps
We study eta invariants of Dirac operators and regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner Müller (see [18], [19]) and use relative traces to define the eta function and the zeta function. We show regularities of eta and zeta functions at the origin so that we can define the eta invariant and the regularized determinant. By the Selberg tra...
متن کاملSpectral Boundary Conditions for Generalizations of Laplace and Dirac Operators
Spectral boundary conditions for Laplace-type operators on a compact manifold X with boundary are partly Dirichlet, partly (oblique) Neumann conditions, where the partitioning is provided by a pseudodifferential projection; they have an interest in string and brane theory. Relying on pseudodifferential methods, we give sufficient conditions for the existence of the associated resolvent and heat...
متن کاملAdiabatic Limits and Foliations
We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with ...
متن کامل