Conceptual questions of quantum gravity and mathematical structures in three-dimensional gravity

到目前为止，引力是四种基本相互作用中唯一一种量子理论无法构建的。这是由于存在现有量化方法无法解决的概念问题。这包括不依赖于固定背景时空的量子理论的构建——在量子引力中，时间和空间本身是要被量子化的——时间在量子理论中的作用，时间和空间是连续的还是离散的问题，以及低能量引力的出现。虽然四维引力的完整量子理论目前还无法实现，但3d引力（一个空间，两个时间维度）作为一个玩具模型，它允许人们在一个完全量子化的理论中解决这些概念问题，并获得与更高维度相关的见解。此外，三维重力在数学中具有很高的重要性。该理论将当前许多领域的研究联系在一起，从而导致该学科取得重大进展。该研究项目旨在解决三维量子引力的几个重要概念问题。这些问题除了对量子引力这一研究课题具有重要意义外，还与数学密切相关。特别是，它们涉及并关联了量子群理论、平面连接的模空间、双曲几何和拓扑量子场论的概念和方法。

Gravity is the only one of the four fundamental interactions for which quantum theory could not be constructed so far. This is due to the presence of conceptual problems which cannot be addressed within existing quantisation approaches. This includes the construction of a quantum theory without relying on a fixed background spacetime - in quantum gravity it is time and space themselves that are to be quantised - the role of time in the quantum theory, the question if time and space are continuous or discrete and the emergence of gravity for low energies. While a full quantum theory of four-dimensional gravity is currently not within reach, 3d gravity (one space, two time dimensions) serves as a toy model which allows one to address these conceptual questions in a fully quantised theory and to gain insights relevant to higher dimensions. Moreover, 3d gravity is of high importance for mathematics. The theory ties together many areas of current research, which lead to major progress in the subject. The research project aims to address several important conceptual questions of 3d quantum gravity. Besides their importance to the research subject of quantum gravity, these questions are also highly relevant to mathematics. In particular, they involve and relate concepts and methods from the theory of quantum groups, moduli spaces of flat connections, hyperbolic geometry and topological quantum field theory.