UC Berkeley

This dissertation explores various aspects of quantization and geometry. In particular, we analyze the ground states of a two-dimensional sigma-model whose target space is an elliptically fibered K3, with the sigma-model compactified on S1 with boundary conditions twisted by a duality symmetry. We show that the Witten index receives contributions from two kinds of states: (i) those that can be mapped to cohomology with coefficients in a certain line bundle over the target space, and (ii) states whose wave-functions are localized at singular fibers. We also discuss the orbifold limit and possible connections with geometric quantization of the target space. We also provide a deformation quantization approach for differential forms on symplectic manifolds. After a description of the Z-graded differential Poisson algebra, we introduce a covariant star product for exterior differential forms and give an explicit expression for it up to second order in the deformation parameter, in the case of symplectic manifolds. The graded differential Poisson algebra endows the manifold with a connection, not necessarily torsion-free, and places upon the connection various constraints.