标题： 四维流形拓扑学，唐纳森-威顿理论，弗洛尔同调以及BV-BFV形式化中的高阶规范理论方法 显示英文标题

标题： 4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism

摘要： 我们研究了基于修正场论设定下四维流形的Donaldson不变量的行为，其中底流形具有边界。众所周知，这些不变量取值于边界三维流形的瞬子Floer同调群中。这些构造的粘合公式根据Atiyah发展的公理系统导出了函子性拓扑场论描述，正如Witten通过超对称Yang-Mills理论的一个版本（即今天所知的Donaldson-Witten理论）所示，这也可以在修正量子场论的框架下考虑。 实际上，可以构建一个AKSZ模型，在特定的规范固定条件下恢复该理论。我们在Cattaneo、Mnev和Reshetikhin最近发展的与切割和粘合兼容的带有边界的流形的修正量子规范形式主义（即BV-BFV形式主义）中考虑这些构造。我们证明了这个理论满足修改后的量子主方程，并在围绕常值背景场扰动时将结果扩展到全局视角。 此外，我们通过BV形式主义处理Donaldson-Witten理论的等变版本，将其与Nekrasov的配分函数联系起来。此外，我们讨论了将其扩展到高阶规范理论以及计数几何方法（如Gromov-Witten和Donaldson-Thomas理论）的可能性，并回顾了它们对于一般Calabi-Yau 3-流形的对应猜想。特别是，我们讨论了相应的（相对）配分函数，即生成给定不变量的生成函数，以及粘合现象。 显示英文摘要

摘要： We study the behavior of Donaldson's invariants of 4-manifolds based on the moduli space of anti self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang-Mills theory, known today as Donaldson-Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. Additionally, we relate these constructions to Nekrasov's partition function by treating an equivariant version of Donaldson-Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov-Witten and Donaldson-Thomas theory and recall their correspondence conjecture for general Calabi-Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.