{"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}
引用次数: 0
Abstract
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.