{"title":"阿贝尔化在Gromov-Witten理论中的一些应用","authors":"Nawaz Sultani, Rachel Webb","doi":"10.1112/jlms.70236","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> be a complex reductive group and let <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>E</mi>\n <annotation>$E$</annotation>\n </semantics></math> be two linear representations of <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math>. Let <span></span><math>\n <semantics>\n <mi>Y</mi>\n <annotation>$Y$</annotation>\n </semantics></math> be a complete intersection in <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> equal to the zero locus of a <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math>-equivariant section of the trivial bundle <span></span><math>\n <semantics>\n <mrow>\n <mi>E</mi>\n <mo>×</mo>\n <mi>X</mi>\n <mo>→</mo>\n <mi>X</mi>\n </mrow>\n <annotation>$E \\times X \\rightarrow X$</annotation>\n </semantics></math>. We explain some general techniques for using quasimap formulas to compute useful <span></span><math>\n <semantics>\n <mi>I</mi>\n <annotation>$I$</annotation>\n </semantics></math>-functions of <span></span><math>\n <semantics>\n <mrow>\n <mi>Y</mi>\n <mrow>\n <mo>/</mo>\n <mo>/</mo>\n </mrow>\n <mi>G</mi>\n </mrow>\n <annotation>$Y\\mathord {/\\hspace{-3.33328pt}/}G$</annotation>\n </semantics></math>. We work several explicit examples, including a rigorous derivation of the conjectural quantum period in Oneto and Petracci (Adv. Geom. <b>18</b> (2018), no. 3, 303–336).</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some applications of abelianization in Gromov–Witten theory\",\"authors\":\"Nawaz Sultani, Rachel Webb\",\"doi\":\"10.1112/jlms.70236\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math> be a complex reductive group and let <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mi>E</mi>\\n <annotation>$E$</annotation>\\n </semantics></math> be two linear representations of <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math>. Let <span></span><math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$Y$</annotation>\\n </semantics></math> be a complete intersection in <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> equal to the zero locus of a <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math>-equivariant section of the trivial bundle <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>E</mi>\\n <mo>×</mo>\\n <mi>X</mi>\\n <mo>→</mo>\\n <mi>X</mi>\\n </mrow>\\n <annotation>$E \\\\times X \\\\rightarrow X$</annotation>\\n </semantics></math>. We explain some general techniques for using quasimap formulas to compute useful <span></span><math>\\n <semantics>\\n <mi>I</mi>\\n <annotation>$I$</annotation>\\n </semantics></math>-functions of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>Y</mi>\\n <mrow>\\n <mo>/</mo>\\n <mo>/</mo>\\n </mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation>$Y\\\\mathord {/\\\\hspace{-3.33328pt}/}G$</annotation>\\n </semantics></math>. We work several explicit examples, including a rigorous derivation of the conjectural quantum period in Oneto and Petracci (Adv. Geom. <b>18</b> (2018), no. 3, 303–336).</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"112 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2025-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70236\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70236","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Some applications of abelianization in Gromov–Witten theory
Let be a complex reductive group and let and be two linear representations of . Let be a complete intersection in equal to the zero locus of a -equivariant section of the trivial bundle . We explain some general techniques for using quasimap formulas to compute useful -functions of . We work several explicit examples, including a rigorous derivation of the conjectural quantum period in Oneto and Petracci (Adv. Geom. 18 (2018), no. 3, 303–336).
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The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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