Symplectic Spinors, Holonomy and Maslov Index
نویسنده
چکیده
In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller ([1]) 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. More precisely, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civitaconnection on L, hence its vanishing 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 endpoint-transverse paths, when compared to the parallel transport w.r.t. to the canonical flat connection of Euclidean space and also for certain elements of the dual spinorbundle along closed or endpoint-transversal paths.
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