Hyperkähler structures in topological field theory

几十年前，Rozansky和Witten为Hyperkähler流形引入了一个新的、部分定义的3D拓扑量子场论，这让数学界感到惊讶。进一步的研究表明，该理论被定义为全纯辛流形，并且扰动不变量给出了全纯辛流形的新特征类。Teleman最近的工作，基于Seiberg和Witten的早期例子，揭示了某些库仑分支的Rozansky-Witten理论和3D中的纯拓扑规范理论之间的二元性。推测，对偶性有一个镜面，它将代数(‘B-模型’)规范理论与由后者的Higgs分支构造的未被发现的3D拓扑规范理论联系在一起，后者似乎确实以扭转Fueter方程的解的形式涉及到Hyperkähler而不仅仅是全纯结构。该项目建议使用代数和解析几何的技术来构建这一理论。Zielinski的主要背景是代数几何和同调代数，特别是在代数几何中使用派生范畴。他的兴趣还包括李代数、代数群和么半群的表示理论。结合起来，这些结构导致了新的代数簇不变量，并可以显示不同类型对象之间令人惊讶的关系。例如，托马斯在本科二年级时承担了第一个暑期阅读项目，以了解贝林森关于射影空间的派生范畴的结构的定理，以及派生范畴中的例外集合与箭图的派生表示之间的一些关系。后来，他还利用Bayer-Macri-Toda的结果研究了Bridgeland稳定性条件，特别是如何在曲线、曲面和猜想上构造这种稳定性条件。更具体地说，该项目集中于了解Bogomolov-Gieseker型不等式和Bridgeland稳定性条件是如何联系的，以及关于派生范畴允许例外集合的变种的稳定性条件的一些构造。这为同伦理论(范畴的局部化、模型结构)以及经典代数几何(如交集理论和特征类)的各种技术提供了很好的培训。这些领域显示出与理论物理的密切相互作用，如拓扑场理论，可以在同调镜像对称方案中观察到。这个项目适合EPSRC几何和拓扑学的研究领域，但也与代数和数学物理的理论方面有关。

Decades ago, Rozansky and Witten surprised the mathematics community by introducing a new, partially defined 3D topological quantum field theory for hyperkähler manifolds. Further study revealed the theory was defined for holomorphic symplectic manifolds, and the perturbative invariants led to new characteristic classes for the latter. Recent work by Teleman, based on early examples by Seiberg and Witten, uncovered a duality between the Rozansky-Witten theory of certain Coulomb branches and pure topological gauge theory in 3D. Conjecturally, the duality has a mirror side which relates algebraic ('B-model') gauge theory with an undiscovered 3D topological gauge theory constructed from Higgs branches of the latter, which does appear to involve the hyperkähler and not just the holomorphic structure, in the form of solutions of a twisted Fueter equation. The project proposes to construct this theory using techniques from algebraic and analytic geometry.Zielinski's main background lies in algebraic geometry and homological algebra, notably in the use of derived categories in algebraic geometry. His interest also include representation theory of Lie algebras and algebraic groups and monoidal categories. In combination, these structures lead to new invariants of algebraic varieties, and can show surprising relationships between different kind of objects. For instance, the first summer reading project Thomas undertook as a second year undergraduate lead to understand Beilinson's theorem about the structure of the derived category of projective space, and some of the relations between exceptional collections in the derived category and derived representations of quivers. Later on, he also studied Bridgeland stability conditions and in particular how to construct such on curves, surfaces and conjecturally on threefolds via results of Bayer-Macri-Toda. More specifically, the project focused on understanding how Bogomolov-Gieseker type inequalities and Bridgeland stability conditions are related, and on some constructions of stability conditions on varieties whose derived category admits an exceptional collection. This has provided good training in the various techniques coming from homotopy theory (localisations of categories, model structures), as well as classical algebraic geometry, such as intersection theory and characteristic classes.These fields show a close interplay with theoretical physics, like topological field theories, as can be observed in the homological mirror symmetry programme. This project fits the EPSRC research areas of Geometry&Topology, but is also relevant to Algebra and the very theoretical side of Mathematical Physics.