Mathematical Foundations of Topological Quantum Field Theories

拓扑量子场论(TQFT)一直是数学与物理之间丰富互动的源泉。这种相互作用在很大程度上依赖于将物理学中的思想转化为严谨的数学环境的能力，从而允许应用强大的定理，如贝兹-多兰柯布尔斯主义假设，这些定理被发现是作为纯粹的数学陈述而发现的。在数学中研究TQFT的最重要的工具是高级范畴理论，特别是完全扩展的边疆范畴。将新的“物理”概念实现为这些领域的数学结构，不仅是理解它们的过程中必要的一步，而且也是作为一门学科研究更高范畴理论的大量例子和动机，这本身就已经很有趣了。在理论凝聚态物理的主题中，TQFT作为低温极限而出现。它们对于在量子计算中使用量子态的拓扑保护的目的特别感兴趣。这类TQFT被分成不同的类别，即所谓的拓扑相。在特定的物理情况下，可以出现哪些拓扑相取决于该情况的参数，主要是维度和对称群G。自那以后，人们对对拓扑相进行分类非常感兴趣，因为给定的参数(G，d)将预测在哪些要求下可以预期特定的相，例如拓扑绝缘体。反射正性(RP)是在大多数物理TQFT中观察到的现象，但到目前为止在数学世界中还没有对应的东西。最近Freed和Hopkins[2]提出了可逆拓扑相的反射正性的定义，并论证了为什么人们应该期望它被应用于所有与物理相关的TQFT。从TQFT的标准定义到rpTQFT的这种改进似乎缩小了从数学模型获得的结果与理论物理文献中出现的结果之间的差距。利用稳定等变同伦理论的工具，他们对最有趣的对称群的可逆rpTQFT进行了低维分类，复制了理论物理文献中已知的结果，并在许多有趣的情况下产生了新的结果。从数学的角度来看，FreedHopkins的RP定义只适用于可逆的TQFT，这是相当不令人满意的。因此，将这个定义扩展到不一定是可逆的TQFT，并将它们的分类结果推广到这种情况是很自然的问题。这个项目主要属于几何和拓扑学的研究领域，但它也与数学和物理的其他几个领域有很强的联系。该研究方法的新颖之处在于将TQFT的物理方面与等变高范畴理论相结合，并在可逆情况下与稳定等变同伦理论相结合。这种互联性在任何一个方向都将是有用的。

The field of Topological Quantum Field Theories (TQFT) has always been a source of rich interaction between Mathematics and Physics. This interplay highly relies on the ability to 'translate' the ideas fromPhysics into a rigorous mathematical setting allowing the application of powerful theorems such as the Baez-Dolan Cobordism Hypothesis, which were discovered as purely mathematical statements. The most important tools for studying TQFT in mathematics are highercategory theory and, in particular, the fully-extended bordism category. Realising new 'physical' ideas as mathematical structures in these areas is not only a necessary step in the process of understanding them, but it also is an ample source of examples and motivation of the study of higher categorytheory as a subject, which is already interesting by itself.In the topic of theoretical condensed matter physics TQFTs arise as lowtemperature limits. They have been of particular interest for the purpose of using topological protection of quantum states in quantum computing.Such TQFTs have been grouped together into different classes, so-called topological phases. Which topological phases can occur in a specific physical situation depends on the parameters of that situation, mainly the dimensiond and the symmetry group G.There has since been a great interest in classifying topological phasesgiven the parameters (G, d) as such a classification would predict underwhich requirements particular phases, such as topological insulators canbe expected.Reflection positivity (rp) is a phenomenon that is observed in most physical TQFTs, but so far had no counter-part in the mathematical world. Recently Freed and Hopkins [2] proposed a definition of reflection positivityfor invertible topological phases and argued why one should expect it to be implemented for all TQFTs relevant in Physics. This improvement from the standard definition of a TQFT to a rpTQFT seems to close the gap between the results obtained from the mathematical model and those appearing in the theoretical physics literature. Using tools from stable equivariant homotopy theory they classified invertible rpTQFTs in low dimensions for most interesting symmetry groups reproducing the results known from the theoretical physics literature and generating new results in many interesting cases. From a mathematical perspective it is quite unsatisfying that the FreedHopkins definition of rp only works out for invertible TQFTs. It is therefore a natural question to ask for this definition to be extended to not necessarily invertible TQFTs and to generalize their classification results to this case.This project mainly falls into the research area of Geometry and Topology, but it also has strong links to several other areas in mathematics and physics. The novelty of the research method lies in combining the physical aspects of TQFT with the theory of equivariant higher categories, and in the invertible case with stable equivariant homotopy theory. This interconnectedness will be useful in either direction.