Symplectic spinors, holonomy and Maslov index

In this note it is shown that the Maslov index for pairs of Lagrangian paths as introduced by Leray and later canonized by Cappell, Lee and Miller appears by parallel transporting elements of ( a certain complex line-subbundle of) the symplectic spinor bundle 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 spinor bundle along closed or endpoint-transversal paths.