scholarly journals Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles

2007 ◽  
Vol 1 (2) ◽  
pp. 305-356 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
James L. Heitsch
Keyword(s):  
Chern Character ◽  
Index Theory ◽  
Dirac Operators ◽  
Type Operator ◽  
Index Formula ◽  
Dirac Type ◽  
Dirac Type Operator ◽  
Index Bundle

AbstractWhen the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.

A Dirac Type Operator on the Non-Commutative Disk

2010 ◽  
Vol 93 (2) ◽  
pp. 107-125 ◽  
Author(s):  
Alan L. Carey ◽  
Sławomir Klimek ◽  
Krzysztof P. Wojciechowski
Keyword(s):  
Type Operator ◽  
Dirac Type ◽  
Dirac Type Operator

INDEX THEORY OF PERTURBED DOLBEAULT OPERATORS: SMOOTH POLAR DIVISORS

2000 ◽  
Vol 11 (02) ◽  
pp. 201-213
Author(s):  
JEFFREY FOX ◽  
PETER HASKELL
Keyword(s):  
Vector Bundles ◽  
Broad Class ◽  
Chern Character ◽  
Index Theory ◽  
Physical Models ◽  
Index Formula ◽  
Complex Manifolds

This paper generalizes the index-theoretic content of the physical models studied in [3, 7]. The paper calculates the homology Chern character, and thus the index formula, for a broad class of perturbed Dolbeault operators on complete noncompact complex manifolds. The manifolds studied are complements of smooth polar divisors of meromorphic sections of vector bundles over closed complex manifolds. These sections define complexes of Koszul type that are used to construct the perturbations of the Dolbeault operators.

scholarly journals An interesting example for spectral invariants

2014 ◽  
Vol 13 (2) ◽  
pp. 305-311 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
James L. Heitsch ◽  
Charlotte Wahl
Keyword(s):  
Chern Character ◽  
Dirac Operators ◽  
Heat Operator ◽  
Spectral Invariants ◽  
Index Bundle

AbstractIn [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals On the index of Dirac operators on arithmetic quotients

1997 ◽  
Vol 56 (3) ◽  
pp. 489-497 ◽  
Author(s):  
Anton Deitmar
Keyword(s):  
Trace Formula ◽  
Special Interest ◽  
Error Term ◽  
Dirac Operators ◽  
Index Formula ◽  
Real Rank ◽  
Present Evidence ◽  
Index Theorems ◽  
Compact Case ◽  
Rational Rank

The aim of this note is to show how the trace formula of Arthur-Selberg can be used to derive index theorems for noncompact arithmetic manifolds. Of special interest is the question, under which circumstances there is an index formula without error term, that is, of the same shape as in the compact case. We shall thus present evidence for the hypothesis that the error term for the Euler operator vanishes in the case that the rational rank is smaller than the real rank.

scholarly journals Higher spectral flow for Dirac operators with local boundary conditions

2016 ◽  
Vol 27 (08) ◽  
pp. 1650068
Author(s):  
Jianqing Yu
Keyword(s):  
Boundary Conditions ◽  
Elliptic Boundary ◽  
Dirac Operators ◽  
Spectral Flow ◽  
Local Boundary ◽  
Manifold With Boundary ◽  
Index Bundle ◽  
Adjoint Extension ◽  
Smooth Map ◽  
Higher Spectral Flow

We consider a one parameter family [Formula: see text] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies [Formula: see text] with [Formula: see text] being a smooth map from the fibration to a unitary group [Formula: see text]. For each [Formula: see text], we impose on [Formula: see text] a certain fixed local elliptic boundary condition [Formula: see text] and get a self-adjoint extension [Formula: see text]. Under the assumption that [Formula: see text] has vanishing [Formula: see text]-index bundle, we establish a formula for the higher spectral flow of [Formula: see text], [Formula: see text]. Our result generalizes a recent result of [A. Gorokhovsky and M. Lesch, On the spectral flow for Dirac operators with local boundary conditions, Int. Math. Res. Not. IMRN (2015) 8036–8051.] to the families case.

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