Symplectic Dirac operators and $${Mp^{\rm c}}$$ -structures

Abstract

Given a symplectic manifold $(M,\omega)$ admitting a metaplectic structure, and choosing a positive $\omega$-compatible almost complex structure $J$ and a linear connection $\nabla$ preserving $\omega$ and $J$, Katharina and Lutz Habermann have constructed two Dirac operators $D$ and ${\wt{D}}$ acting on sections of a bundle of symplectic spinors. They have shown that the commutator $[ D, {\wt{D}}]$ 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 $\Mpc$ structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in $H^2(M,\Z)$. For any $\Mpc$ 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 $\mathcal{P}$ 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 $\mathcal{P}$ stabilizes the sections of each of those.