{"title":"量子仿射w -代数的模不变性","authors":"Victor G. Kac, Minoru Wakimoto","doi":"10.1007/s00220-024-05223-z","DOIUrl":null,"url":null,"abstract":"<div><p>We find modular transformations of normalized characters for the following <i>W</i>-algebras: (a) <span>\\(W_k^{min}(\\mathfrak {g}), \\text {where } \\mathfrak {g}=D_n (n\\ge 4), \\text {or } E_6, E_7, E_8,\\)</span> and <i>k</i> is a negative integer <span>\\(\\ge -2\\)</span>, or <span>\\(\\ge -\\frac{h^\\vee }{6}-1\\)</span>, respectively; (b) quantum Hamiltonian reduction of the <span>\\(\\hat{\\mathfrak {g}}\\)</span>-module <span>\\(L(k \\Lambda _0)\\)</span>, where <span>\\(\\mathfrak {g}\\)</span> is a simple Lie algebra, <i>f</i> is its non-zero nilpotent element, and <i>k</i> is a principal admissible level with the denominator <span>\\(u>\\theta (x)\\)</span>, where 2<i>x</i> is the Dynkin characteristic of <i>f</i>, and <span>\\(\\theta \\)</span> is the highest root of <span>\\(\\mathfrak {g}\\)</span>. We prove that these vertex algebras are modular invariant. A conformal vertex algebra <i>V</i> is called modular invariant if its character <span>\\(tr_V q^{L_0-c/24}\\)</span> converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of <i>V</i> is important since, in particular, conjecturally it implies that <i>V</i> is simple, and that <i>V</i> is rational, provided that it is lisse.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 2","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Modular Invariance of Quantum Affine W-Algebras\",\"authors\":\"Victor G. Kac, Minoru Wakimoto\",\"doi\":\"10.1007/s00220-024-05223-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We find modular transformations of normalized characters for the following <i>W</i>-algebras: (a) <span>\\\\(W_k^{min}(\\\\mathfrak {g}), \\\\text {where } \\\\mathfrak {g}=D_n (n\\\\ge 4), \\\\text {or } E_6, E_7, E_8,\\\\)</span> and <i>k</i> is a negative integer <span>\\\\(\\\\ge -2\\\\)</span>, or <span>\\\\(\\\\ge -\\\\frac{h^\\\\vee }{6}-1\\\\)</span>, respectively; (b) quantum Hamiltonian reduction of the <span>\\\\(\\\\hat{\\\\mathfrak {g}}\\\\)</span>-module <span>\\\\(L(k \\\\Lambda _0)\\\\)</span>, where <span>\\\\(\\\\mathfrak {g}\\\\)</span> is a simple Lie algebra, <i>f</i> is its non-zero nilpotent element, and <i>k</i> is a principal admissible level with the denominator <span>\\\\(u>\\\\theta (x)\\\\)</span>, where 2<i>x</i> is the Dynkin characteristic of <i>f</i>, and <span>\\\\(\\\\theta \\\\)</span> is the highest root of <span>\\\\(\\\\mathfrak {g}\\\\)</span>. We prove that these vertex algebras are modular invariant. A conformal vertex algebra <i>V</i> is called modular invariant if its character <span>\\\\(tr_V q^{L_0-c/24}\\\\)</span> converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of <i>V</i> is important since, in particular, conjecturally it implies that <i>V</i> is simple, and that <i>V</i> is rational, provided that it is lisse.</p></div>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":\"406 2\",\"pages\":\"\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2025-01-18\",\"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://link.springer.com/article/10.1007/s00220-024-05223-z\",\"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://link.springer.com/article/10.1007/s00220-024-05223-z","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}
On Modular Invariance of Quantum Affine W-Algebras
We find modular transformations of normalized characters for the following W-algebras: (a) \(W_k^{min}(\mathfrak {g}), \text {where } \mathfrak {g}=D_n (n\ge 4), \text {or } E_6, E_7, E_8,\) and k is a negative integer \(\ge -2\), or \(\ge -\frac{h^\vee }{6}-1\), respectively; (b) quantum Hamiltonian reduction of the \(\hat{\mathfrak {g}}\)-module \(L(k \Lambda _0)\), where \(\mathfrak {g}\) is a simple Lie algebra, f is its non-zero nilpotent element, and k is a principal admissible level with the denominator \(u>\theta (x)\), where 2x is the Dynkin characteristic of f, and \(\theta \) is the highest root of \(\mathfrak {g}\). We prove that these vertex algebras are modular invariant. A conformal vertex algebra V is called modular invariant if its character \(tr_V q^{L_0-c/24}\) converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of V is important since, in particular, conjecturally it implies that V is simple, and that V is rational, provided that it is lisse.
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.