4D TQFT and categorified Hall algebras

量子场论是用来描述和研究基本粒子的工具。它是最成功和最精确的科学理论之一，具有广泛的应用：从粒子加速器到超导体和量子计算机。量子场依赖于空间和时间的基本几何。在某些情况下，这些场与长度和持续时间的测量无关，只取决于空间和时间的基本形状。在这种情况下，量子场论称为拓扑场论。拓扑量子场论（TQFTs）是当前物理学和数学研究的前沿。TQFT的数学构造是基于公理的集合，很像，例如，欧几里得几何。这使得TQFT成为一种数学和物理理论。TQFT是由物理学家在80年代后期发明的，目的是为了建立一个严格的量子场论。他们很快被理解为广泛的数学兴趣，被有关，除其他外，结理论和理论的四流形代数拓扑，理论的模空间代数几何和量子群。近三十年来，TQFT一直是数学和理论物理的中心研究领域之一。唐纳森、琼斯、维滕和孔采维奇都因在拓扑场论方面的工作获得了菲尔兹奖章。然而，理论的实质部分，特别是推广到高维的情况下，仍然有待建设。这是本计画的最终目标，我们提出的方法是建立一个连贯的架构，将TQFT方法中的几个最新结果加以整合。其中一个主要的工具，目前在我们的处置是高等范畴理论，这是一个新的和非常活跃的领域的纯数学。分类方法背后的基本思想是，在最抽象的层面上阐述问题往往会导致新的见解和创造全新的解决方案。正因为如此，范畴语言在纯数学中越来越普遍，甚至在一些更应用的学科领域，如化学和计算机科学。高级范畴理论试图进一步发展这种语言，以便将其应用于数学的其他领域，通常更具有几何风味，包括TQFT。我们方法中的第二个重要对象是量子群。量子群最初与TQFT大约在同一时间被定义。它们最初被认为是代数对象，也就是说，它们的定义涉及公式和方程。很快，人们发现它们与数学的不同领域，特别是TQFT，有许多令人惊讶的深刻联系。在这一主题的几个早期有影响力的著作中，量子群被证明可以产生三维流形的不变量，即纯几何性质的对象。通过这种方式，量子群显然包含了丰富的几何信息。为了构造四维流形的不变量，我们必须把量子群的定义再深一层，定义一个具有额外信息层的类似对象。这样的过程被称为分类。我们打算利用量子群引入以来积累的大量研究成果和来自高等范畴理论的新思想来这样做。这将使我们能够开发一种系统的方法来构建四维TQFT以及构建这些理论的新的和具体的例子。最终的结果将是产生一个算法，为四维对象分配各种数字，表达其深层内在属性。

Quantum field theory is the tool used to describe and study fundamental particles. It is one of the most successful and precise scientific theories with a wide range of applications: from particle accelerators to superconductors and quantum computers. Quantum fields depend on the underlying geometry of space and time. In some cases the fields are independent of the measurment of length and duration and only depend on the underlying shape of space and time. In such cases the quantum field theory is called topological. Topological Quantum Field Theories (TQFTs) are now in the frontier of research in both physics and mathematics. The mathematical construction of TQFTs is based on a collection of axioms, much like, for example, Euclidean geometry. This makes TQFTs a mathematical as well as physical theory. TQFTs were invented by physicists in the late 80's in an effort to make a mathematicallly rigorous construction of Quantum Field Theories. They were quickly understood to be of wide mathematical interest, being related to, among other things, Knot Theory and the theory of four-manifolds in Algebraic Topology, the theory of moduli spaces in algebraic geometry and Quantum Groups. For almost three decades TQFT has been one of the central research areas in mathematics and theoretical physics. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. However substantial parts of the theory, in particular the generalization to the higher-dimensional case, still remain to be constructed. This is the ultimate goal of the proposed project.The way we propose to achieve it is to create a coherent framework unifying several recent results in the approach to TQFT. One of the main tools currently at our disposal is Higher Category Theory which is a novel and very active area of pure mathematics. The fundamental idea behind the categorical approach is that formulating the problems on the most abstract level often leads to new insights and creation of radically new approaches to their solutions. For this reason the categorical language is becoming more and more widespread in Pure Mathematics and even in some areas of more applied disciplines, such as Chemistry and Computer Science. Higher Category Theory seeks to develop this language further in order to apply it to additional areas of mathematics, generally speaking of a more geometric flavor, including TQFT. The second important object in our approach are Quantum Groups. Quantum groups were originally defined at around the same time as TQFTs. They were originally conceived as algebraic objects, that is their definition involved formulas and equations. Very soon they were found to have a number of surprising and deep connections to different areas of mathematics, in particular to TQFT. In several early influential works on the subject, Quantum Groups were shown to give rise to invariants of three-dimensional manifolds, i.e. objects of a purely geometric nature. In this way it became apparent that Quantum Groups contain a rich stock of geometric information. To construct invariants of four-dimensional manifolds we must take this definition of Quantum Groups one level deeper, defining a similar object with an additional layer of information. Such a process is called categorification. We intend to do so using the wealth of research on Quantum Groups accumulated since their introduction and the new ideas coming from Higher Category Theory. This should then allow us to develop a systematic approach to constructing four-dimensional TQFTs as well as constructing new and concrete examples of these theories. The ultimate result would be to produce an algorithm assigning various numbers to a four-dimensional object, expressing it's deep intrinisic properties.