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Symplectic Dirac operators and Mpc -structures
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Cahen, M., Gutt, Simone and Rawnsley, John H. (2011) Symplectic Dirac operators and Mpc -structures. General Relativity and Gravitation, Vol.43 (No.12). pp. 3593-5617. doi:10.1007/s10714-011-1239-x ISSN 0001-7701.
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Official URL: http://dx.doi.org/10.1007/s10714-011-1239-x
Abstract
Given a symplectic manifold (M, ω) admitting a metaplectic structure, and choosing a positive ω-compatible almost complex structure J and a linear connection preserving ω and J, Katharina and Lutz Habermann have constructed two Dirac operators D and D acting on sections of a bundle of symplectic spinors. They have shown that the commutator [DD] is an elliptic operator preserving an infinite number of finite dimensional subbundles. We extend the construction of symplectic Dirac operators to any symplectic manifold, through the use of Mpc structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in H2(MZ) . For any Mpc structure, choosing J and a linear connection as before, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator is elliptic. Using the Fock description of the spinor space allows the definition of a notion of degree and the construction of a dense family of finite dimensional subbundles; the operator stabilizes the sections of each of those.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Dirac equation, Symplectic manifolds, Spinor analysis | ||||
Journal or Publication Title: | General Relativity and Gravitation | ||||
Publisher: | Springer | ||||
ISSN: | 0001-7701 | ||||
Official Date: | December 2011 | ||||
Dates: |
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Volume: | Vol.43 | ||||
Number: | No.12 | ||||
Page Range: | pp. 3593-5617 | ||||
DOI: | 10.1007/s10714-011-1239-x | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Date of first compliant deposit: | 17 December 2015 | ||||
Date of first compliant Open Access: | 17 December 2015 | ||||
Funder: | Communauté française de Belgique. Action de Recherche Concertée |
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