On Mpc-structures and Symplectic Dirac Operators

TL;DR

This paper investigates the properties of symplectic Dirac operators on compact symplectic manifolds, computes their kernels in specific cases, and explores the structure of Mpc-structures and related symmetry groups.

Contribution

It establishes finite dimensionality of kernels, computes these kernels for complex projective spaces, and classifies G-invariant Mpc-structures with a new variant of Parthasarathy's formula.

Findings

Kernels of symplectic Dirac operators are finite dimensional on compact manifolds.

Explicit kernel computations for complex projective spaces.

Classification of G-invariant Mpc-structures and a new commutator formula.

Abstract

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group Mpc and classify G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space.