{"title":"对数)交织算子的模块不变性","authors":"Yi-Zhi Huang","doi":"10.1007/s00220-024-04983-y","DOIUrl":null,"url":null,"abstract":"<p>Let <i>V</i> be a <span>\\(C_2\\)</span>-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-<i>q</i>-traces (<span>\\(q=e^{2\\pi i\\tau }\\)</span>) shifted by <span>\\(-\\frac{c}{24}\\)</span> of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized <i>V</i>-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-<i>q</i>-traces were proved by Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) and Fiordalisi (Commun Contemp Math 18:1650026, 2016) using the method developed in Huang (Commun Contemp Math 7:649–706, 2005). The method that we use to prove this conjecture is based on the theory of the associative algebras <span>\\(A^{N}(V)\\)</span> for <span>\\(N\\in \\mathbb {N}\\)</span>, their graded modules and their bimodules introduced and studied by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) and Huang (Commun Math Phys 396:1–44, 2022). This modular invariance result gives a construction of <span>\\(C_2\\)</span>-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Modular Invariance of (Logarithmic) Intertwining Operators\",\"authors\":\"Yi-Zhi Huang\",\"doi\":\"10.1007/s00220-024-04983-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>V</i> be a <span>\\\\(C_2\\\\)</span>-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-<i>q</i>-traces (<span>\\\\(q=e^{2\\\\pi i\\\\tau }\\\\)</span>) shifted by <span>\\\\(-\\\\frac{c}{24}\\\\)</span> of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized <i>V</i>-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-<i>q</i>-traces were proved by Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) and Fiordalisi (Commun Contemp Math 18:1650026, 2016) using the method developed in Huang (Commun Contemp Math 7:649–706, 2005). The method that we use to prove this conjecture is based on the theory of the associative algebras <span>\\\\(A^{N}(V)\\\\)</span> for <span>\\\\(N\\\\in \\\\mathbb {N}\\\\)</span>, their graded modules and their bimodules introduced and studied by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) and Huang (Commun Math Phys 396:1–44, 2022). This modular invariance result gives a construction of <span>\\\\(C_2\\\\)</span>-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-05-14\",\"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/s00220-024-04983-y\",\"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/s00220-024-04983-y","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
让 V 是一个没有非零负重元素的 \(C_2\)- 无限顶点算子代数。我们证明了这样一个猜想:由等级受限的广义 V 模块间几何修正(对数)交织算子乘积的伪 q 迹((q=e^{2\pi i\tau }))的解析广延(\(-\frac{c}{24}\)移动)所跨越的空间在模量变换下是不变的。Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) 和 Fiordalisi (Commun Contemp Math 18:1650026, 2016) 使用 Huang (Commun Contemp Math 7:649-706, 2005) 中开发的方法证明了提出这一猜想所需的收敛性和解析扩展结果,以及关于这种移位伪 Q 迹的一些后果。我们用来证明这个猜想的方法是基于作者在 Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) 和 Huang (Commun Math Phys 396:1-44, 2022) 中介绍和研究的 \(N\in \mathbb {N}\) 的关联代数(A^{N}(V)\)、它们的分级模块和它们的双模的理论。这个模块不变性结果给出了从相应的零属对数共形场论构造\(C_2\)-无限属一对数共形场论的方法。
Modular Invariance of (Logarithmic) Intertwining Operators
Let V be a \(C_2\)-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-q-traces (\(q=e^{2\pi i\tau }\)) shifted by \(-\frac{c}{24}\) of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized V-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-q-traces were proved by Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) and Fiordalisi (Commun Contemp Math 18:1650026, 2016) using the method developed in Huang (Commun Contemp Math 7:649–706, 2005). The method that we use to prove this conjecture is based on the theory of the associative algebras \(A^{N}(V)\) for \(N\in \mathbb {N}\), their graded modules and their bimodules introduced and studied by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) and Huang (Commun Math Phys 396:1–44, 2022). This modular invariance result gives a construction of \(C_2\)-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.
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.