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It is shown that the Maslov Index for pairs of Lagrangian Paths as introducedby Cappell, Lee and Miller appears by parallel transporting elements of (acertain complex line-subbundle of) the symplectic spinorbundle over Euclideanspace, when pulled back to an (embedded) Lagrangian submanifold $L$, alongclosed or non-closed paths therein. More precisely, the CLM-Index mod 4determines the holonomy group of this line bundle w.r.t. theLevi-Civita-connection on $L$, hence its vanishing is equivalent to theexistence of a trivializing parallel section. Moreover, it is shown that theCLM-Index determines parallel transport in that line-bundle along arbitraryendpoint-transverse paths, when compared to the parallel transport w.r.t. tothe canonical flat connection of Euclidean space and also for certain elementsof the dual spinor-bundle along closed or endpoint-transversal paths.

Symplectic spinors, holonomy and Maslov index