Symplectic Spinors, Holonomy and Maslov Index

TL;DR

This paper links the Maslov Index for Lagrangian paths to the holonomy of a symplectic spinor bundle, revealing how the CLM-Index mod 4 governs parallel transport and triviality of the bundle.

Contribution

It establishes a novel connection between the Maslov Index and the holonomy of symplectic spinor bundles, providing new insights into their geometric and topological properties.

Findings

The CLM-Index mod 4 determines the holonomy group of the spinor line bundle.

Vanishing CLM-Index mod 4 implies the existence of a trivializing parallel section.

The CLM-Index controls parallel transport along paths with specific endpoint conditions.

Abstract

In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along closed or non-closed paths therein. In especially, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on $L$, hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown that the CLM-Index determines parallel transport in that line-bundle along arbitrary paths when compared to the parallel transport w.r.t. to the canonical flat connection of Euclidean space, if the Lagrangian tangent planes at the endpoints either coincide or are orthogonal. This is derived from a result on parallel transport of certain elements of the dual spinorbundle along closed or endpoint-transversal paths.