Mathematical Foundations of Topological Quantum Field Theories

拓扑量子场论（TQFT）领域一直是数学和物理之间丰富互动的源泉。这种相互作用高度依赖于将物理学中的思想“转化”为严格的数学设置的能力，从而允许应用强大的定理，如贝兹-多兰协同假设，这些定理是作为纯数学陈述发现的。在数学中研究TQFT最重要的工具是高范畴理论，特别是完全扩展的边界范畴。在这些领域中，将新的“物理”思想作为数学结构来实现，不仅是理解它们的必要步骤，而且也是研究高等范畴论的一个充足的例子来源和动机，这本身就很有趣。在理论凝聚态物理中，tqft是作为低温极限出现的。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。这样的tqft被归类为不同的类别，即所谓的拓扑相。在特定的物理情况下，哪些拓扑相可以发生取决于该情况的参数，主要是维度和对称群G。从那时起，对给定参数（G, d）的拓扑相分类产生了极大的兴趣，因为这样的分类可以预测哪些需要特定的相，例如拓扑绝缘体可以预期。反射正性（rp）是在大多数物理tqft中观察到的一种现象，但到目前为止，在数学世界中还没有相应的现象。最近Freed和Hopkins提出了可逆拓扑相位反射正性的定义，并讨论了为什么人们应该期望它适用于所有与物理学相关的tqft。从TQFT的标准定义到rpTQFT的改进似乎缩小了从数学模型得到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，他们对大多数有趣的对称群的低维可逆rptqft进行了分类，再现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tqft是非常不令人满意的。因此，要求将该定义扩展到不一定可逆的tqft并将其分类结果推广到这种情况是一个自然的问题。本项目主要属于几何和拓扑的研究领域，但与数学和物理的其他几个领域也有很强的联系。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，在可逆情况下与稳定等变同伦理论相结合。这种相互联系在任何一个方向上都是有用的。

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.