Groupoids and Geometric Quantization

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The mathematical models underlying classical mechanics and quantum mechanics are very different. Indeed, classical mechanics describes physical laws on a large scale (e.g. laws of motion for a swinging pendulum, planetary orbits, etc.) and is mathematically rooted in differential geometry, an extension of calculus to more general spaces that allow for curvature and higher dimensions. In contrast, quantum mechanics describes the physical laws on a subatomic scale and is rooted in the language of Hilbert spaces. There is great interest in understanding the connection between these two models. Intuitively, one expects the laws describing the physics at the subatomic level determine behaviour on the large scale. Mathematically, this intuition is realized by a process known as the semi-classical limit, which produces a classical system from a quantum one. Proceeding in the reverse direction (i.e. finding a quantum system that corresponds to a classical one) is known as quantization, although it is not always possible for physical and mathematical reasons. Geometric quantization is a mathematical understanding of such a relationship between classical and quantum physics. Being geometric, it highlights symmetries that are present in each framework. This research proposal investigates ways in which the geometric approach to quantization can be adapted to certain kinds of generalized symmetries, known as groupoid actions, a modern perspective on symmetry at the crossroads of geometry, topology, and mathematical physics. Like their classical counterparts, these symmetries can be viewed as transformations of the parameters in the mathematical models that do not change the physical laws being described. In contrast to the classical setting, groupoid symmetries cannot always be composed: it is not always possible to follow a given transformation by another one. The particular groupoid symmetries investigated in this proposal, called quasi-symplectic groupoid actions, describe examples of interest, such as moduli spaces of connections (which can be viewed as spaces of solutions of a differential equation), from various active areas of mathematics research in Canada and abroad, including symplectic geometry, algebraic geometry, and conformal and quantum field theories. A complete understanding of quantization for groupoid actions is not yet available.  The objectives in this research proposal contribute significantly towards filling this gap by (i) advancing the theoretical framework that accommodates generalized symmetry; and (ii) working within one such a framework (quasi-Hamiltonian actions) to ultimately clarify active research in physics (e.g. by proving conjectured formulas in conformal field theory).