| Subelliptic boundary conditions for SpinC-Dirac operators, gluing, relative indices, and tame Fredholm pairs. | |
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MedLine Citation:
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PMID: 17032764 Owner: NLM Status: PubMed-not-MEDLINE |
Abstract/OtherAbstract:
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Let X be a Spin manifold with boundary, such that the Spin structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szego projectors, we define modified partial differential-Neumann boundary conditions, Reo, for spinors, which lead to subelliptic Fredholm boundary value problems for the Spin-Dirac operator, eth(eo). To study the index of these boundary value problems we introduce a generalization of Fredholm pairs to the "tame" category. In this context, we show that the index of the graph closure of (eth(eo), Reo) equals the tame relative index, on the boundary, between Reo and the Calderon projector. Let X0 and X1 be strictly pseudoconvex, Spin manifolds, as above. Let phi : bX1 --> bX0, be a contact diffeomorphism, S0, S1 denote generalized Szego projectors on bX0, bX1, respectively, and R0(eo), R1(eo), the subelliptic boundary conditions they define. If X1 is the manifold X1 with its orientation reversed, then the glued manifold X = X0 coproduct operator(phi) X1 has a canonical Spin structure and Dirac operator, ethX(eo). Applying these results we obtain a formula for the relative index, R-Ind(S0, phi*S1), [formula: see text]. As a special case, this formula verifies a conjecture of Atiyah and Weinstein [(1997) RIMS Kokyuroku 1014:1-14] for the index of the quantization of a contact transformation between cosphere bundles. |
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Authors:
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Charles L Epstein |
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Publication Detail:
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Type: Journal Article; Research Support, Non-U.S. Gov't; Research Support, U.S. Gov't, Non-P.H.S. Date: 2006-10-10 |
Journal Detail:
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Title: Proceedings of the National Academy of Sciences of the United States of America Volume: 103 ISSN: 0027-8424 ISO Abbreviation: Proc. Natl. Acad. Sci. U.S.A. Publication Date: 2006 Oct |
Date Detail:
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Created Date: 2006-10-18 Completed Date: 2007-01-26 Revised Date: 2008-11-20 |
Medline Journal Info:
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Nlm Unique ID: 7505876 Medline TA: Proc Natl Acad Sci U S A Country: United States |
Other Details:
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Languages: eng Pagination: 15364-9 Citation Subset: - |
Affiliation:
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Department of Mathematics and Radiology, University of Pennsylvania, Philadelphia, PA 19104, USA. cle@math.upenn.edu |
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From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine
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