Hyperkähler structures in topological field theory

几十年前，Rozansky和维滕为超凯勒流形引入了一个新的、部分定义的3D拓扑量子场论，这让数学界大吃一惊。进一步的研究表明，该理论被定义为全纯辛流形，微扰不变量导致后者的新的特征类。Teleman最近的工作，基于Seiberg和维滕早期的例子，揭示了某些库仑分支的Rozansky-维滕理论和3D纯拓扑规范理论之间的对偶性。猜想上，对偶性有一个镜像的一面，它将代数（“B模型”）规范理论与一个未被发现的三维拓扑规范理论联系起来，后者是由后者的希格斯分支构建的，它似乎确实涉及超凯勒而不仅仅是全纯结构，以扭曲的富特方程的解的形式。该项目提出，以构建这一理论使用的技术，从代数和解析几何。Zielinski的主要背景在于代数几何和同调代数，特别是在使用派生类别代数几何。他的兴趣还包括代表理论的李代数和代数群和monoidal类别。结合起来，这些结构导致代数簇的新不变量，并且可以显示不同种类对象之间的惊人关系。例如，第一个夏季阅读项目托马斯承诺作为第二年的本科导致了解贝林森定理的结构派生类别的射影空间，以及一些关系之间的特殊集合在派生类别和派生表示的箭袋。后来，他还研究了Bridgeland稳定性条件，特别是如何建设这样的曲线，曲面和constructurally对threefolds通过结果的拜尔马克里户田。更具体地说，该项目侧重于了解Bogomolov-Gieseker型不等式和Bridgeland稳定性条件是如何相关的，以及在其派生范畴允许例外集合的品种上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（如相交理论和特征类）的各种技术提供了良好的训练。这些领域与理论物理（如拓扑场论）有着密切的相互作用，这在同调镜像对称计划中可以观察到。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.