{"title":"适当朗道-金兹堡势能、内在镜像对称性和相对镜像图","authors":"Fenglong You","doi":"10.1007/s00220-024-04954-3","DOIUrl":null,"url":null,"abstract":"<p>Given a smooth log Calabi–Yau pair (<i>X</i>, <i>D</i>), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (<i>X</i>, <i>D</i>). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When <i>D</i> is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety <i>X</i>, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When <i>X</i> is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map\",\"authors\":\"Fenglong You\",\"doi\":\"10.1007/s00220-024-04954-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given a smooth log Calabi–Yau pair (<i>X</i>, <i>D</i>), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (<i>X</i>, <i>D</i>). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When <i>D</i> is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety <i>X</i>, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When <i>X</i> is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-03-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s00220-024-04954-3\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-04954-3","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
摘要
给定光滑对数 Calabi-Yau 对 (X,D),我们利用本征镜像对称构造定义镜像适当朗道-金兹堡势,并证明它是 (X,D) 的两点相对格罗莫夫-维滕不变式的生成函数。我们计算了具有多个负接触阶的某些相对不变式,然后应用范等人的相对镜像定理(Sel Math (NS) 25(4):Art.54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z)计算两点相对不变式。当 D 是 nef 时,我们计算适当的 Landau-Ginzburg 势,并证明它是相对镜像映射的逆。将其特殊化到环综 X 的情况下,这意味着 m Gräfnitz 等人(2022 年)的猜想,即适当的朗道-金兹堡势是开放镜像映射。当 X 是法诺变时,适当的势与正则量子周期的反求有关。
The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map
Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.