caa a22 4500 445386320 CHVBK 20180317143033.0 cr unu---uuuuu 170323e20111201xx s 000 0 eng 10.1007/s10714-011-1239-x doi (NATIONALLICENCE)springer-10.1007/s10714-011-1239-x Symplectic Dirac operators and $${Mp^{\rm c}}$$ -structures [Elektronische Daten] [Michel Cahen, Simone Gutt, John Rawnsley] Given a symplectic manifold (M, ω) admitting a metaplectic structure, and choosing a positive ω-compatible almost complex structure J and a linear connection $${\nabla}$$ preserving ω and J, Katharina and Lutz Habermann have constructed two Dirac operators D and $${{\tilde{D}}}$$ acting on sections of a bundle of symplectic spinors. They have shown that the commutator $${[ D, {\tilde{D}}]}$$ 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 $${Mp^{\rm c}}$$ structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in $${H^2(M, {\mathbb Z})}$$ . For any $${Mp^{\rm c}}$$ structure, choosing J and a linear connection $${\nabla}$$ as before, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator $${\fancyscript{P}}$$ 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 $${\fancyscript{P}}$$ stabilizes the sections of each of those. Springer Science+Business Media, LLC, 2011 Symplectic spinors nationallicence Dirac operators nationallicence $${Mp^{\rm c}}$$ structures nationallicence Cahen Michel Département de Mathématique, Université Libre de Bruxelles, Campus Plaine, C. P. 218, Boulevard du Triomphe, 1050, Bruxelles, Belgium aut Gutt Simone Département de Mathématique, Université Libre de Bruxelles, Campus Plaine, C. P. 218, Boulevard du Triomphe, 1050, Bruxelles, Belgium aut Rawnsley John Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK aut General Relativity and Gravitation Springer US; http://www.springer-ny.com 43/12(2011-12-01), 3593-3617 0001-7701 43:12<3593 2011 43 10714 https://doi.org/10.1007/s10714-011-1239-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10714-011-1239-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Cahen Michel Département de Mathématique, Université Libre de Bruxelles, Campus Plaine, C. P. 218, Boulevard du Triomphe, 1050, Bruxelles, Belgium aut NATIONALLICENCE 700 1- Gutt Simone Département de Mathématique, Université Libre de Bruxelles, Campus Plaine, C. P. 218, Boulevard du Triomphe, 1050, Bruxelles, Belgium aut NATIONALLICENCE 700 1- Rawnsley John Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK aut NATIONALLICENCE 773 0- General Relativity and Gravitation Springer US; http://www.springer-ny.com 43/12(2011-12-01), 3593-3617 0001-7701 43:12<3593 2011 43 10714 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer