The Index of Dirac Operators on Incomplete Edge Spaces

Albin, Pierre; Gell-Redman, Jesse

doi:10.3842/sigma.2016.089

Cited by 11 publications

(32 citation statements)

References 64 publications

(115 reference statements)

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“…The spin Dirac operator on a model edge space is indeed a generalized Dirac operator in the sense that it is given by the differential expression (4.1) and satisfies the commutator relations (4.3). This has been established by Albin and Gell-Redman [AlGR16]. In this subsection we prove that the Gauss-Bonnet operator on a model edge space is a generalized Dirac operator in the sense above as well.…”

Section: Examples Of Generalized Dirac Operators On An Abstract Edgesupporting

confidence: 61%

On the domain of Dirac and Laplace type operators on stratified spaces

2018

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We consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric. Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces. This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces. Our argument does not rely on microlocal techniques and is very explicit. The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales. Our results hold for the Gauss-Bonnet and spin Dirac operators satisfying a spectral Witt condition.

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Section: Examples Of Generalized Dirac Operators On An Abstract Edgesupporting

confidence: 61%

On the domain of Dirac and Laplace type operators on stratified spaces

2018

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“…Assume for the moment that the higher order term h in the Riemannian metric g on M is zero. Then the explicit structure of D is studied in [AlGe13].…”

Section: Microlocal Analysis Of the Heat Kernel On Edgesmentioning

confidence: 99%

“…Singular analysis has been employed by Bismut and Cheeger [BiCh90] in their families index theorem on manifolds with boundary, where they assumed invertibility of the boundary Dirac operators. Let us also mention work by Brüning [Bru09] for the signature operator on simple edge spaces of Witt type and the work by Albin and Gell-Redman [AlGe13] for the spin Dirac operator on simple edge spaces satisfying a geometric Witt condition (see also the recent contribution [AlGe17]); in these articles an explicit index formula is proved.…”

mentioning

confidence: 99%

Eta and rho invariants on manifolds with edges

Piazza¹,

Vertman²

2019

Annales De l'Institut Fourier

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We establish existence of eta-invariants as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature and the spin Dirac operator. We derive an analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments employ microlocal analysis of the heat kernel asymptotics on incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As an application, we discuss stability results for the two rho-invariants we have defined.Date: This document was compiled on: December 14, 2018. 2000 Mathematics Subject Classification. 58J20, 58J28. 1 2. Review of geometry on incomplete edge spaces Consider a compact smoothly stratified space M of depth 1 and dimension m: we shall assume that M is the disjoint union of the top stratum M, a smooth manifold of dimension m, and finitely many lower dimensional strata {B i }, i ∈ I, where each B i is a closed compact manifold of dimension b i . For notational simplicity we continue with the case of a single stratum B of dimension b, the general case is studied in an analogous way. The stratification hypothesis asserts the existence of an open neighbourhood U ⊂ M around the singular stratum B, together with a distance function x : U → [0, 1), such that U ∩ M is the total space of a smooth fibre bundle over B with the fibre given by a truncated cone C (F ) = (0, 1) × F over a compact smooth manifold F of dimension f ≥ 1. The distance function x restricts to a radial function of that cone on each fibre. We shall refer to such a stratified space as a simple edge space, or, shortly, as an edge space.The stratified space M can be resolved to define a compact manifold M c , with boundary ∂M c being the total space of a fibration φ : ∂M c → B with fibre F . Under the resolution, the neighborhood U lifts to a collar neighborhood U ⊂ M c , which is a smooth fibration of cylinders [0, 1) × F over B with radial function x. The open regular stratum M is identified with M c \∂M c .Notation: with a small abuse of notation we shall denote ∂M c by ∂M.An edge structure on M is defined by a particular choice of a Riemannian metric. 7 the bordisms are assumed to have edges of odd dimension.

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“…For β = 1, P 2 β (C) has an edge cone singularity of cone angle 2π β along S 2 ∞ , hence the index theorem for closed manifolds needs to be modified to take into account the conical singularity. Such an extension has been considered in [4], where the following result is proved: Let X be a spin oriented 4-manifold, Y be a smooth compact oriented embedded surface, g an incomplete edge metric on X \ Y with cone angle 2π β along Y . Then, for β ∈ (0, 1],…”

Section: The Aps Index Theorem For M βmentioning

confidence: 99%

Harmonic spinors on a family of Einstein manifolds

Franchetti

2018

Nonlinearity

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The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and P 2 (C) with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of C 2 and extend to P 2 (C) as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah-Patodi-Singer index theorem need to be carefully considered in order to show agreement between the index of the twisted Dirac operator and the result obtained by counting the explicit solutions.

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The Index of Dirac Operators on Incomplete Edge Spaces

Cited by 11 publications

References 64 publications

On the domain of Dirac and Laplace type operators on stratified spaces

On the domain of Dirac and Laplace type operators on stratified spaces

Eta and rho invariants on manifolds with edges

Harmonic spinors on a family of Einstein manifolds

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