Hyperkähler structures in topological field theory

拓扑场论中的超克勒结构

• 批准号: 1941556
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1941556
• 关键词: Hyperk; hler; structures; topological; field

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.

几十年前，Rozansky和Witten为hyperkähler流形引入了一个新的、部分定义的三维拓扑量子场论，震惊了数学界。进一步的研究表明，该理论是为全纯辛流形定义的，并且微扰不变量导致了后者的新特征类。Teleman最近的工作，基于Seiberg和Witten早期的例子，揭示了三维中某些库仑分支的Rozansky-Witten理论和纯拓扑规范理论之间的对偶性。从推测上讲，对偶有一面镜像面，它将代数（“b模型”）规范理论与由后者的希格斯分支构建的未被发现的三维拓扑规范理论联系起来，这似乎涉及hyperkähler，而不仅仅是全纯结构，以扭曲Fueter方程的解的形式。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构代数，特别是在代数几何中派生范畴的使用。他的兴趣还包括李代数的表示理论、代数群和一元范畴。结合起来，这些结构导致代数变量的新不变量，并可以显示不同类型对象之间的惊人关系。例如，Thomas在大学二年级的第一个暑期阅读项目中理解了Beilinson关于射影空间派生范畴结构的定理，以及派生范畴中的异常集合与颤栗的派生表示之间的一些关系。后来，他还研究了桥地稳定性条件，特别是如何根据Bayer-Macri-Toda的结果在曲线、曲面和猜想三倍上构造桥地稳定性条件。更具体地说，该项目侧重于了解Bogomolov-Gieseker型不等式和bridgeeland稳定性条件之间的关系，以及其派生类别允许特殊集合的品种的稳定性条件的一些构造。这为来自同伦理论（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）的各种技术提供了良好的训练。这些场显示出与理论物理密切的相互作用，如拓扑场论，可以在同调镜像对称程序中观察到。该项目适合EPSRC的几何与拓扑研究领域，但也与代数和数学物理的理论方面相关。