Fully extended r-spin topological field theories

该项目是纯数学-拓扑场论（TFTs），高等范畴理论和代数几何-和理论物理的接口。我会用物理学的概念来证明数学定理。我计划在双范畴和（无穷大，2）范畴设置下研究具有切向结构的二维（2d）完全扩展泛函tft：由矩阵分解描述的Landau-Ginzburg （LG）模型的拓扑b -扭转和其他提供某些微分梯度（dg）类别的拓扑扭曲sigma模型。在此过程中，我将应用和发展LG /（分数）Calabi-Yau （CY）对应，a -∞范畴和圆作用在更高范畴范畴上的同伦不动点的新结果。该项目的主要目标是通过考虑不同的目标双类别和（无穷，2）类别，在r-自旋表面上找到完全扩展的2d泛函tft的有趣例子。我想了解分数CY品种的作用-与某些类别的矩阵分解相关-作为LG模型的目标。另一个令人兴奋的例子来源应该来自与物理学中sigma模型的拓扑扭曲相关的dg类别。在完全扩展tft的双范畴公式中，可以应用已证明的协点假设，二维旋转群的同伦不动点的计算，对矩阵分解的双范畴的很好的理解，以及已知的g-范畴的2-可对偶性的条件。对于项目的第二部分，我需要构建一个（无穷，2）-类别的矩阵分解，我将考虑无穷类别中的代数，特别是在dg-类别的dg-神经中，作为框架和r-自旋tft的新目标。该项目结合了过去几年可用的最先进的结果，以产生2d完全扩展r-自旋tft的新例子。寻找维度高于1的例子是一项重要的任务，因为尽管存在分类猜想（以及双范畴公式中的定理），但在文献中几乎没有这样的例子。本课题的研究成果将为拓扑场理论和高等范畴理论提供重要的见解，并将为理论物理中拓扑扭曲sigma模型的严格数学表述迈出一大步。

The project is at the interface of pure mathematics – topological field theories (TFTs), higher category theory and algebraic geometry – and theoretical physics. I will use ideas from physics to prove theorems in mathematics.I plan to work on 2-dimensional (2d) fully extended functorial TFTs with tangential structure in the bicategorical and in the (infinity,2)-categorical setting: topological B-twist of Landau-Ginzburg (LG) models described by matrix factorisations and other topologically twisted sigma models that provide certain differential graded (dg) categories.On the way I will apply and develop new results on the LG / (fractional) Calabi-Yau (CY) correspondence, A-infinity categories and homotopy fixed points of circle actions on higher categories categories.The main objective of the project is to find interesting examples of fully extended 2d functorial TFTs on r-spin surfaces by considering different target bicategories and (infinity,2)-categories. I would like to understand the role of fractional CY varieties – related to certain categories of matrix factorisations – as targets of LG models. Another exciting source of examples should come from dg-categories related to topological twists of sigma models in physics.In the bicategorical formulation of fully extended TFTs the proven Cobordism Hypothesis, the computation of the homotopy fixed points of the 2d rotational group, the good understanding of the bicategory of matrix factorisations, and the known conditions on 2-dualisability of dg-categories can be applied.For the second part of the project I need to construct an (infinity,2)-category of matrix factorisations, I will consider algebras in infinity categories, in particular in the dg-nerve of a dg-category to serve as new targets for framed and r-spin TFTs. The project combines state-of-the-art results that became available in the past years to produce new examples of 2d fully extended r-spin TFTs. It is an important task to find such examples in dimension higher than 1 as – although classification conjectures (and theorems in the bicategorical formulation) exist – examples are almost absent in the literature.The result of this project should deliver important insights in topological filed theory and higher category theory and would make a big step towards rigorous mathematical formulation of topologically twisted sigma models in theoretical physics.