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Please use this identifier to cite or link to this item: http://hdl.handle.net/1807/11205

Title: Spin-c Quantization, Prequantization and Cutting
Authors: Fuchs, Shay
Advisor: Karshon, Yael
Department: Mathematics
Keywords: Quantization
Symplectic Geometry
Issue Date: 31-Jul-2008
Abstract: In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.
URI: http://hdl.handle.net/1807/11205
Appears in Collections:Doctoral
Department of Mathematics - Doctoral theses

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