The cyclic open–closed map, u-connections and R-matrices

Kai Hugtenburg
{"title":"The cyclic open–closed map, u-connections and R-matrices","authors":"Kai Hugtenburg","doi":"10.1007/s00029-024-00925-7","DOIUrl":null,"url":null,"abstract":"<p>This paper considers the (negative) cyclic open–closed map <span>\\({\\mathcal{O}\\mathcal{C}}^{-}\\)</span>, which maps the cyclic homology of the Fukaya category of a symplectic manifold to its <span>\\(S^1\\)</span>-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that <span>\\({\\mathcal{O}\\mathcal{C}}^{-}\\)</span> intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to <span>\\({\\mathcal{O}\\mathcal{C}}^{-}\\)</span> in the semisimple case; we also consider the non-semisimple case.\n</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"128 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Selecta Mathematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00029-024-00925-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

This paper considers the (negative) cyclic open–closed map \({\mathcal{O}\mathcal{C}}^{-}\), which maps the cyclic homology of the Fukaya category of a symplectic manifold to its \(S^1\)-equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that \({\mathcal{O}\mathcal{C}}^{-}\) intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to \({\mathcal{O}\mathcal{C}}^{-}\) in the semisimple case; we also consider the non-semisimple case.

Abstract Image

循环开闭图、u 连接和 R 矩阵
本文考虑了(负)循环开闭映射({\mathcal{O}\mathcal{C}}^{-}\),它将交错流形的 Fukaya 范畴的循环同调映射到其\(S^1\)-等变量子同调。我们证明(在简化的技术假设下)这一映射在等变参数方向上尊重各自的自然连接。在单调设置中,这让我们得出结论:\({\mathcal{O}\mathcal{C}}^{-}\) 将量子杯积的特征值与第一切尔恩类的 Fukaya 范畴分解,与量子同调的 Hukuhara-Levelt-Turrittin 分解交织在一起。我们还解释了我们的结果与半简单同调场论的 Givental-Teleman 分类的关系:特别是,在半简单情况下,R 矩阵与 \({mathcal{O}\mathcal{C}}^{-}\) 的关系;我们还考虑了非半简单情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信