Singular topological field theory and classifying spaces of derived manifolds

奇异拓扑场论和导出流形的空间分类

• 批准号: 19K14522
• 负责人: Macpherson Andrew
• 金额: $ 1万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Early-Career Scientists
• 财政年份: 2019
• 资助国家: 日本
• 起止时间: 2019-04-01 至 2020-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-19K14522/
• 关键词: Virtual cycles; topological field theory; bordism; derived geometry; orientations; Topological field theory; Derived geometry; Bordism

In my research plan I proposed a project whose objective was to assign cycles of integration in bordism theory to derived manifolds with tangential structures (such as orientations, spin structures, and so on), enhancing the theory of "virtual cycles" developed for use in Gromov-Witten theory and also Spivak's bordism theory of unoriented derived manifolds. Results in this direction would be a stepping stone to defining "enhanced" Gromov-Witten type invariants, with many conceivable applications.I succeeded in proving the first main statement outlined in the application, that is, that families of derived manifolds with tangential structures are classified by a well-known object, a "Thom spectrum." This means that if a moduli space can be given the structure of a derived manifold --- something which is the case in many important examples --- it can be assigned a cycle in a bordism ring, which can be thought of as a more structured version of the notion of "counting points." Hence, this work is likely to have ramifications in Floer theory, symplectic field theory, and beyond.I spoke on my results at an international conference in Osaka in November. Since the grant terminated early, I did not have time to write up and publish the arguments, so the paper remains in draft form. I continue to work on it and aim to publish before the grant would have terminated given its full length, next February.

在我的研究计划中，我提出了一个项目，其目标是将边界论中的积分循环分配给具有切向结构(如方向、自旋结构等)的派生流形，从而增强了为用于Gromov-Witten理论和斯皮瓦克的无定向派生流形的边界主义理论而开发的“虚拟循环”理论。这个方向的结果将是定义“增强型”Gromov-Witten类型不变量的垫脚石，具有许多可以想象的应用。我成功地证明了应用程序中概述的第一个主要陈述，即具有切向结构的派生流形的族由一个众所周知的对象分类，即“Thom谱”。这意味着，如果一个模空间可以被赋予派生流形的结构-在许多重要的例子中就是这样-它可以被分配到一个有界环中的一个圈，这可以被认为是一个更结构化的“计点”概念的版本。因此，这项工作可能会对弗洛尔理论、辛场理论等产生影响。11月，我在大阪的一个国际会议上谈到了我的结果。由于拨款提前终止，我没有时间写出和发表论点，所以论文仍然是草稿形式。我还在继续写这本书，我的目标是在明年2月这笔拨款终止之前发表。

项目成果

The operad that corepresents enrichment

核心呈现丰富性的操作

• 发表时间: 2019
• 期刊: arXiv
• 作者: Kitamura;M.;Sasaki;K.;Ishii;T.;& Watanabe;K.;Andrew W.Macpherson
• 通讯作者: Andrew W.Macpherson

Symmetries of enriched higher category theory

丰富的高范畴论的对称性

• 发表时间: 2019
• 作者: Kitamura;M.;Sasaki;K.;Ishii;T.;& Watanabe;K.;Andrew W.Macpherson;土谷昭善;大矢 浩徳;佐々木恭志郎・朱思斉・姜月・錢コン・山田祐樹;Andrew W.Macpherson
• 通讯作者: Andrew W.Macpherson

Field theory, derived geometry, and virtual cycles

场论、导出几何和虚拟循环

• 发表时间: 2019
• 作者: Kitamura;M.;Sasaki;K.;Ishii;T.;& Watanabe;K.;Andrew W.Macpherson;土谷昭善;大矢 浩徳;佐々木恭志郎・朱思斉・姜月・錢コン・山田祐樹;Andrew W.Macpherson;Akiyoshi Tsuchiya;大矢 浩徳;佐々木恭志郎・米満文哉・山田祐樹;Andrew W.Macpherson
• 通讯作者: Andrew W.Macpherson