Cahen, M. (Michel), 1935-, Gutt, Simone and Rawnsley, John H. (John Howard), 1947-. (2011) Symplectic Dirac operators and Mpc -structures. General Relativity and Gravitation, Vol.43 (No.12). pp. 3593-5617. ISSN 0001-7701

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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
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of 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
Date:	December 2011
Volume:	Vol.43
Number:	No.12
Page Range:	pp. 3593-5617
Identification Number:	10.1007/s10714-011-1239-x
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Communauté française de Belgique. Action de Recherche Concertée
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URI:	http://wrap.warwick.ac.uk/id/eprint/37975

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