UC Berkeley

In the thesis, we initial first steps in understanding Quantum Mirror Symmetry and noncommutative compactification of moduli spaces of tori. To obtain a global invariant of noncommutative torus bundle, we study the monodromy of Gauss-Manin connection on periodic cyclic homology groups of Heisenberg group. Then the global monodromy map is developed, and provides a important criterion to detect when a noncommutative torus bundle is dequantizable. A process to construct the dequantizing Poisson manifold is given, when the dequantization criterion is satisfied. Naively, it seems that the Morita theory for noncommutative torus bundles can be developed naturally as in Rieffel theory for rotation algebras. However, this assumption turns out to be wrong; the Morita class of a non-dequantizable noncommutative torus bundle is not a classical object in the category of C*- algebras, and we call them C*- stacks. Even with the extended notion of C*- stacks, the Morita theory is still incompatible with noncommutative torus bundles with the infinite Poisson limit. There is no rotation algebra that can be used to compactify the moduli space of rotation algebras, even though we know that the "infinite Poisson rotation algebra" is strongly Morita equivalent to the classical torus. This subtlety is completely solved by a new mathematical structure hidden behind the quantization of constant Dirac structures on the tori.

We develop a new theory of quantum spaces called spatial structure to give a better understanding of quantum spaces and torus fibrations. We clarify some examples of spatial algebras and develop a rigorous way to construct a monoidal structure from the spatial structure. Using Hilsum-Skandalis maps between groupoids, we find that a groupoid presentation of a C* - algebra implies a monoidal structure on the category of representations. We decompose the spatial product of the cyclic modules over the rotation algebras as an example, and propose a conjecture that a quantum mirror symmetry lies behind the spatial structure and the Hopfish structure in the sense of Tang and Weinstein.