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Spinc, Mpc and symplectic Dirac operators

par Cahen, Michel ;Gutt, Simone
Référence New Trends in Mathematics Teaching, 61, page (13-28)
Publication Publié, 2013

Article révisé par les pairs

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Résumé :

We advertise the use of the group Mpc (a circle extension of the symplectic group) instead of the metaplectic group (a double cover of the symplectic group). The essential reason is that Mpc-structures exist on any symplectic manifold. They first appeared in the framework of geometric quantization [4, 10]. In a joint work with John Rawnsley [1], we used them to extend the definition of symplectic spinors and symplectic Dirac operators which were first introduced by Kostant [9] and K. Habermann [6] in the presence of a metaplectic structure. We recall here this construction, stressing the analogies with the group Spinc in Riemannian geometry. Dirac operators are defined as a contraction of Clifford multiplication and covariant derivatives acting on spinor fields; in Riemannian geometry, the contraction is defined using the Riemannian structure. In symplectic geometry one contracts using the symplectic structure or using a Riemannian structure defined by the choice of a positive compatible almost complex structure. We suggest here more general contractions yielding new Dirac operators.