Details
| Original language | English |
|---|---|
| Pages (from-to) | 220-252 |
| Number of pages | 33 |
| Journal | Journal des Mathematiques Pures et Appliquees |
| Volume | 120 |
| Early online date | 3 Aug 2017 |
| Publication status | Published - Dec 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.
Keywords
- Groupoid, Index theory, Lie manifold
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Journal des Mathematiques Pures et Appliquees, Vol. 120, 12.2018, p. 220-252.
Research output: Contribution to journal › Article › Research › 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 -