{"title":"Virasoro Constraints for Toric Bundles","authors":"Tom Coates, Alexander Givental, Hsian-Hua Tseng","doi":"10.1017/fmp.2024.2","DOIUrl":null,"url":null,"abstract":"<p>We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203073427923-0125:S2050508624000027:S2050508624000027_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$E \\to B$</span></span></img></span></span> if and only if it holds for the base <span>B</span>. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of <span>E</span>, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of <span>B</span>, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.</p>","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.8000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum of Mathematics Pi","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fmp.2024.2","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.
我们证明了格罗莫夫-维滕理论中的维拉索罗猜想（Virasoro conjecture）在环束 $E \to B$ 的总空间中成立，前提是且仅当它在基 B 中成立时：(i) 我们建立了一个局部化公式，用环状纤维的零属不变式和 B 的全属不变式来表达 E 的格罗莫夫-维滕不变式，相对于纤维环状作用等变，以及 (ii) 我们利用布朗关于环状束的镜像定理来传递这个公式中的非变极限。
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$E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.
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