Groupoids and Geometric Quantization

经典力学和量子力学背后的数学模型是非常不同的。事实上，经典力学描述了大范围的物理定律(例如摆动摆的运动定律、行星轨道等)。它在数学上植根于微分几何，这是微积分在更一般的空间中的扩展，允许曲率和更高的维度。相比之下，量子力学在亚原子尺度上描述物理定律，植根于希尔伯特空间的语言。人们对理解这两个模型之间的联系非常感兴趣。 直觉上，人们期待在亚原子水平上描述物理的定律决定大规模的行为。在数学上，这种直觉是通过一个被称为半经典极限的过程实现的，这个过程从量子系统产生经典系统。以相反的方向前进(即找到与经典量子系统相对应的量子系统)被称为量子化，尽管由于物理和数学原因并不总是可能的。 几何量子化是对经典物理和量子物理之间这种关系的数学理解。由于是几何的，它突出了每个框架中存在的对称性。这项研究方案研究了如何使量子化的几何方法适用于某些类型的广义对称，即所谓的群组作用，以及从几何学、拓扑学和数学物理的十字路口对对称性的现代观点。 就像它们的经典对称一样，这些对称可以被视为数学模型中参数的变换，而不会改变所描述的物理定律。与经典的背景相反，群胚对称并不总是可以合成的：在一个给定的变换之后跟随另一个变换并不总是可能的。在这一建议中研究的特殊的群胚对称，称为准辛群作用，描述了感兴趣的例子，例如来自加拿大和国外数学研究的各个活跃领域的连接的模空间(可以被视为微分方程解的空间)，包括辛几何、代数几何、共形和量子场论。 对于群体性行为的量化还没有一个完整的理解。本研究提案中的目标通过(I)推进适应广义对称性的理论框架；以及(Ii)在一个这样的框架内工作(准哈密顿作用)，最终澄清物理学中的活跃研究(例如，通过证明保形场论中的猜想公式)，从而大大有助于填补这一空白。

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).