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U. Bunke

Publications and source records attributed to U. Bunke.

15 recordsLinked to original sources
arXiv2006

Index theory, eta forms, and Deligne cohomology

The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lott's index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher generalizations. Our main technical tool is a variant of local index theory for Dirac operators of families of manifolds with corners.

math.DG↗
arXiv2003

Determinant bundles, boundaries, and surgery

In this note we specialize and illustrate the ideas developed in the paper math.DG/0201112 of the first author ("Index theory, 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 obtained by the second author and K. Wojciechowski.

math.DG↗
arXiv2002

Homotopy quantum field theory and the index gerbe

Given a family of Dirac operators with vanishing spectral flow we construct a thin-invariant rank-one field theory in the sense of Turner and Willerton arXiv:math.AT/0201116. Our construction of the field theory generalizes the one of the index gerbe by Lott, arXiv:math.DG/0106177, and it also complements the relation between gerbes and thin-invariant rank-one field theories studied in arXive:math.AT/0201116.

math.DG↗
arXiv2000

Towards the trace formula for convex-cocompact groups

We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this subgroup is an invariant distribution on the group, and we state precise conjecture about its Fouriertransform. We apply our trace formula to resolvent kernels. This yields functional equations of Selberg zeta functions and information about their singularities. The paper leaves open several interesting questions concerning the proof of the conjecture mentioned above on the one hand (this has to do with estimating the growth of the Eisenstein series as the spectral parameter turns to infinity), and the integrality of the multiplicities and of certain residues on the other hand.

math.DG↗
arXiv1999

On the index of equivariant Toeplitz operators

The goal is to understand the index-theoretic aspects of the recent preprint of R. Nest and F. Radulescu, math.OA/9911042. The basic observation (due to E. Guenter/N. Higson) is that the index of the Toeplitz operator is equal to the index of an associated Callias type operator, i.e. a Dirac operator with potential, the index of which is easy to compute. We show how to extend this idea to the equivariant case.

math.DG↗
arXiv1998

The spectrum of Kleininan manifolds

We obtain the Plancherel theorem for the quotient of a simple Lie group of real rank one by a convex-cocompact discrete subgroup and its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. We develop a geometric version of scattering theory. The paper is an update of dg-ga/9609011, which is completely rewritten from the point of view of representation theory. In particular, we are now able to show the meromorphy of Eisenstein series for all parameters.

math.DG↗
arXiv1997

Higher analytic torsion and cohomology of diffeomorphism groups

We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural flat connection and a vertical Riemannian metric. The higher analytic torsion form of Bismut/Lott associated to the situation is invariant with respect to the connected component of the identity of the diffeomorphism group of $M$. Using that the space of Riemannian metrics is contractible we define continuous cohomology classes of the diffeomorphism group and its Lie algebra. For the circle we compute this classes in degree 2 and show that the group cohomology class is non-trivial, while the Lie algebra cohomology class vanishes.

dg-ga↗
arXiv1997

The eta-form and a generalized Maslov index

We consider families of Dirac operators on the unit interval which depend on parameters via boundary conditions. We study the associated eta forms and Maslov cocyles. With this simple example we show how previous results of Lesch/Woiciechowski and the first author on the eta invariant of cylinders generalize to the family case.

dg-ga↗
arXiv1996

The spectrum of Kleinian manifolds

A Kleinian manifold Y is a quotient of a rank-one symmetric space of non-compact type by a convex-cocompact discrete group of isometries. We describe the spectral decomposition of the space of square integrable sections of locally homogeneous bundles on Y with respect to locally invariant differential operators. In the course of the proof we obtain meromorphic continuations of Eisenstein series and scattering matrices.

dg-ga↗
arXiv1995

Cohomological properties of the smooth globalization of a Harish-Chandra module

We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these globalizations for the nilpotent part of the Iwasawa decomposition of the group.

math.RT↗