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，该项目的目的是将Bordism理论中的整合循环分配给具有切向结构（例如方向，旋转结构等）的流形，从而增强了用于Gromov-Witten理论的“虚拟循环”理论，以及Spivak spivak spivak spivak spivak of specivak specivak of specivak of specivak specivak of specivak of specivak of specivak of specivak of specivak of dienteriented sprientient偏离的偏见。在这个方向上的结果将是定义“增强” Gromov-Witten类型不变性的垫脚石，并具有许多可疑的应用。我成功证明了应用程序中概述的第一个主要陈述，即，与切向结构的派生流形的家属由一个知名的对象分类为“ Thom Spectrum”。这意味着，如果可以给出模量空间的结构，那么在许多重要示例中，它就是这种情况 - 可以将其分配给边界环中的周期，可以将其视为“计数点”概念的更结构化版本。因此，这项工作很可能在浮子理论，符号田地理论等方面产生了影响。我在11月在大阪举行的国际会议上谈到了我的结果。自从赠款提早终止以来，我没有时间写并发布论点，因此该论文仍以草稿形式。我将继续努力并打算在明年2月的全长情况下终止赠款之前发表。

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

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

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