University of Toronto's Symplectic Geometry Seminar

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. Guillemin, Sternberg, and Weitsman define in their paper the concept of \emph{Signature Quantization} and show that it is \emph{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 $(M,\omega)$, a spin$^c$-prequantization is a spin$^c$-structure together with a connection compatible with $\omega$. We explain how one can cut a spin$^c$-prequantization and show that the choice of a spin$^c$-structure on $\mathbb C$ (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.