{"title":"计算 $${\\mathbb {C}\\mathbb {P}}^1$ 上的微分函数","authors":"Alexandr Buryak, Paolo Rossi","doi":"10.1007/s11005-024-01823-x","DOIUrl":null,"url":null,"abstract":"<div><p>We give explicit formulas for the number of meromorphic differentials on <span>\\(\\mathbb{C}\\mathbb{P}^1\\)</span> with two zeros and any number of residueless poles and for the number of meromorphic differentials on <span>\\(\\mathbb{C}\\mathbb{P}^1\\)</span> with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on <span>\\(\\mathbb{C}\\mathbb{P}^1\\)</span> with vanishing residue conditions at a subset of poles, up to the action of <span>\\(\\textrm{PGL}(2,\\mathbb {C})\\)</span>. The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of <span>\\(\\textrm{SL}_2(\\mathbb {C})\\)</span> and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 4","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01823-x.pdf","citationCount":"0","resultStr":"{\"title\":\"Counting meromorphic differentials on \\\\({\\\\mathbb {C}\\\\mathbb {P}}^1\\\\)\",\"authors\":\"Alexandr Buryak, Paolo Rossi\",\"doi\":\"10.1007/s11005-024-01823-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We give explicit formulas for the number of meromorphic differentials on <span>\\\\(\\\\mathbb{C}\\\\mathbb{P}^1\\\\)</span> with two zeros and any number of residueless poles and for the number of meromorphic differentials on <span>\\\\(\\\\mathbb{C}\\\\mathbb{P}^1\\\\)</span> with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on <span>\\\\(\\\\mathbb{C}\\\\mathbb{P}^1\\\\)</span> with vanishing residue conditions at a subset of poles, up to the action of <span>\\\\(\\\\textrm{PGL}(2,\\\\mathbb {C})\\\\)</span>. The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of <span>\\\\(\\\\textrm{SL}_2(\\\\mathbb {C})\\\\)</span> and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.</p></div>\",\"PeriodicalId\":685,\"journal\":{\"name\":\"Letters in Mathematical Physics\",\"volume\":\"114 4\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-07-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11005-024-01823-x.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Letters in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11005-024-01823-x\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Letters in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11005-024-01823-x","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Counting meromorphic differentials on \({\mathbb {C}\mathbb {P}}^1\)
We give explicit formulas for the number of meromorphic differentials on \(\mathbb{C}\mathbb{P}^1\) with two zeros and any number of residueless poles and for the number of meromorphic differentials on \(\mathbb{C}\mathbb{P}^1\) with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on \(\mathbb{C}\mathbb{P}^1\) with vanishing residue conditions at a subset of poles, up to the action of \(\textrm{PGL}(2,\mathbb {C})\). The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of \(\textrm{SL}_2(\mathbb {C})\) and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.
期刊介绍:
The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.