Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 220-252 |
| Seitenumfang | 33 |
| Fachzeitschrift | Journal des Mathematiques Pures et Appliquees |
| Jahrgang | 120 |
| Frühes Online-Datum | 3 Aug. 2017 |
| Publikationsstatus | Veröffentlicht - Dez. 2018 |
Abstract
We prove an index theorem of Atiyah–Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Allgemeine Mathematik
- Mathematik (insg.)
- Angewandte Mathematik
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in: Journal des Mathematiques Pures et Appliquees, Jahrgang 120, 12.2018, S. 220-252.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Getzler rescaling via adiabatic deformation and a renormalized index formula
AU - Bohlen, Karsten
AU - Schrohe, Elmar
N1 - Publisher Copyright: © 2017 Elsevier Masson SAS Copyright: Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2018/12
Y1 - 2018/12
N2 - We prove an index theorem of Atiyah–Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.
AB - We prove an index theorem of Atiyah–Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.
KW - Groupoid
KW - Index theory
KW - Lie manifold
UR - http://www.scopus.com/inward/record.url?scp=85028318542&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1607.07039
DO - 10.48550/arXiv.1607.07039
M3 - Article
AN - SCOPUS:85028318542
VL - 120
SP - 220
EP - 252
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
SN - 0021-7824
ER -