The symplectic Dirac and Dolbeault operators and the Lichnerowicz Laplacian

Abstract

The Laplacian operator $\mathcal{P}$ arising as a commutator of two symplectic Dirac operators introduced in [1] in the context of Schrödinger picture and rediscovered in [2] as a commutator of symplectic Dirac-Dolbeault operators in the Fock picture admits a natural geometric interpretation, which is described in this thesis.

One part of the thesis deals with the central extension of the metaplectic group $Mp$ and its maximal compact subgroup $MU^c$. We discuss the existence and classification of $Mp-$ and $\Mp^c-$structures on almost symplectic manifolds and describe the correspondence with $Spin-$ resp. $Spin^c-$structures. After that we introduce Laplacians $\Delta^{1,0}$ and $\Delta^{0,1}$ on the complex symmetric algebra arising from the Lichnerowicz Laplacian $\Delta$ after the complexification of the tangent bundle and complex bilinear extension of the canonical Hermitian connection, when considering almost Kähler manifolds. Having explained the reason, why the symplectic spinor bundle can be regarded as the tensor bundle $L\otimes S^{*,0}(M)$, for some Hermitian line bundle $L$ over $M$, we adjust appropriately the symplectic Clifford multiplication and give then an interpretation of the symplectic Dirac-Dolbeault operators, regarding a symplectic manifold as almost Kähler equipped with the canonical Hermitian connection, and prove the main theorem, which can be stated as follows. Theorem: If the line bundle $L$ is trivial, the operator $\mathcal{P}$ coincides with $-\Delta^{1,0}$.