{"title":"Crossed modular categories and the Verlinde formula for twisted conformal blocks","authors":"Tanmay Deshpande, S. Mukhopadhyay","doi":"10.4310/cjm.2023.v11.n1.a2","DOIUrl":null,"url":null,"abstract":"In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $\\Gamma$ and a positive integral level $\\ell$ under the assumption that \"$\\Gamma$ preserves a Borel\". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $\\Gamma$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $\\Gamma$-crossed modular functor and show that it is very closely related to the notion of a $\\Gamma$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $\\Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $\\Gamma$-crossed modular functor. Along the way, we prove equivalence between a $\\Gamma$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cambridge Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cjm.2023.v11.n1.a2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 8
Abstract
In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $\Gamma$ and a positive integral level $\ell$ under the assumption that "$\Gamma$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $\Gamma$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $\Gamma$-crossed modular functor and show that it is very closely related to the notion of a $\Gamma$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $\Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $\Gamma$-crossed modular functor. Along the way, we prove equivalence between a $\Gamma$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.