Symplectic Dirac Operators and Mpc-structures

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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 [D, 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 structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in H²(M,). For any 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 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 P stabilizes the sections of each of those.