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

到目前为止，在四种基本相互作用中，引力是唯一一种无法构建量子理论的。这是因为存在一些概念上的问题，而这些问题在现有的量化方法中是无法解决的。这包括在不依赖固定的时空背景的情况下构建量子理论--在量子引力中，要量化的是时间和空间本身--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。虽然四维引力的全量子理论目前还不是触手可及的，但三维重力(一个空间，两个时间维度)就像一个玩具模型，允许人们在完全量化的理论中解决这些概念性问题，并获得与更高维度相关的见解。此外，3D重力对数学具有很高的重要性。这一理论将当前研究的许多领域联系在一起，导致了该学科的重大进展。该研究项目旨在解决3D量子引力的几个重要概念问题。除了它们对量子引力研究主题的重要性外，这些问题也与数学高度相关。特别是，它们涉及并关联了量子群论、平连通模空间、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.