Zusammenfassung
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 resealing technique similar in spirit to Getzler's resealing. With a given Lie manifold we associate an appropriate ...
Zusammenfassung
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 resealing technique similar in spirit to Getzler's resealing. 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 resealed bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold. (C) 2017 Elsevier Masson SAS. All rights reserved.
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