The index of Callias-type operators with Atiyah–Patodi–Singer boundary conditions

Shi, Pengshuai

doi:10.1007/s10455-017-9575-z

Cited by 8 publications

(4 citation statements)

References 27 publications

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“…Bär and Ballmann, [6,7], showed that an elliptic boundary value problem for a Callias-type operator on a complete manifold with compact boundary is Fredholm and studied its index. P. Shi, [37], proved a version of the Callias-type index theorem for the Atiyah-Patodi-Singer (APS) boundary value problem for Callias-type operators on a complete manifold with compact boundary.…”

Section: Introductionmentioning

confidence: 99%

The Atiyah–Patodi–Singer Index on Manifolds with Non-compact Boundary

Braverman

Shi

2020

J Geom Anal

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We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold M . We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds N0 and N1. If the dimension of M is odd we show that the latter index depends only on the restrictions A0 and A1 of D to N0 and N1 and thus is an invariant of the boundary. We use this invariant to define the relative η-invariant η(A1, A0). We show that even though in our situation the η-invariants of A1 and A0 are not defined, the relative η-invariant behaves as if it were the difference η(A1) − η(A0).

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Section: Introductionmentioning

confidence: 99%

The Atiyah–Patodi–Singer Index on Manifolds with Non-compact Boundary

Braverman

Shi

2020

J Geom Anal

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“…P. Shi, [37], proved a version of the Callias-type index theorem for the APS boundary value problem for Callias-type operators on a complete odd-dimensional manifold with compact boundary.…”

Section: Introductionmentioning

confidence: 99%

APS index theorem for even-dimensional manifolds with non-compact boundary

Braverman¹,

Shi²

2017

Preprint

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We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold M . We use this index to define the relative ηinvariant η(A1, A0) of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the η-invariants of A1 and A0 are not defined, the relative η-invariant behaves as if it were the difference η(A1) − η(A0). We also define the spectral flow of a family of such operators and use it to compute the variation of the relative η-invariant.

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“…On noncompact manifolds, APS index theory of Callias-type Dirac operators was developed by Braverman and Shi [14,15,55,56]. Braverman also obtained a result for Callias-type operators on noncompact Lorentzian manifolds with boundary [12], which was proved in the compact case by Bär-Strohmaier [6].…”

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confidence: 99%

An equivariant Atiyah-Patodi-Singer index theorem for proper actions I: the index formula

Hochs¹,

Wang²,

Wang³

2019

Preprint

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Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac type operator on M under a suitable boundary condition has an equivariant index in the Ktheory of the reduced group C * -algebra C * r G of G, which is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. Using a trace on a suitable subalgebra of C * r G, defined by the orbital integral over the conjugacy class of an element g ∈ G, we prove an equivariant Atiyah-Patodi-Singer index theorem in this setting. We first state general analytic conditions under which this theorem holds, and then show that these conditions are satisfied if g = e is the identity element; if G is a finitely generated, discrete group, and the conjugacy class of g has polynomial growth; and if G is a connected, linear, real semisimple Lie group, and g is a semisimple element. In the classical case, where M is compact and G is trivial, our arguments reduce to a relatively short and simple proof of the original Atiyah-Patodi-Singer index theorem. Contents

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The index of Callias-type operators with Atiyah–Patodi–Singer boundary conditions

Cited by 8 publications

References 27 publications

The Atiyah–Patodi–Singer Index on Manifolds with Non-compact Boundary

The Atiyah–Patodi–Singer Index on Manifolds with Non-compact Boundary

APS index theorem for even-dimensional manifolds with non-compact boundary

An equivariant Atiyah-Patodi-Singer index theorem for proper actions I: the index formula

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