{"title":"Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules","authors":"Bin Gui","doi":"10.1007/s00220-024-04959-y","DOIUrl":null,"url":null,"abstract":"<p>A unitary and strongly rational vertex operator algebra (VOA) <span>\\({\\mathbb {V}}\\)</span> is called strongly unitary if all irreducible <span>\\({\\mathbb {V}}\\)</span>-modules are unitarizable. A strongly unitary VOA <span>\\({\\mathbb {V}}\\)</span> is called completely unitary if for each unitary <span>\\({\\mathbb {V}}\\)</span>-modules <span>\\({\\mathbb {W}}_1,{\\mathbb {W}}_2\\)</span> the canonical non-degenerate Hermitian form on the fusion product <span>\\({\\mathbb {W}}_1\\boxtimes {\\mathbb {W}}_2\\)</span> is positive. It is known that if <span>\\({\\mathbb {V}}\\)</span> is completely unitary, then the modular category <span>\\(\\textrm{Mod}^\\textrm{u}({\\mathbb {V}})\\)</span> of unitary <span>\\({\\mathbb {V}}\\)</span>-modules is unitary (Gui in Commun Math Phys 372(3):893–950, 2019), and all simple VOA extensions of <span>\\({\\mathbb {V}}\\)</span> are automatically unitary and moreover completely unitary (Gui in Int Math Res Not 2022(10):7550–7614, 2022; Carpi et al. in Commun Math Phys 1–44, 2023). In this paper, we give a geometric characterization of the positivity of the Hermitian product on <span>\\({\\mathbb {W}}_1\\boxtimes {\\mathbb {W}}_2\\)</span>, which helps us prove that the positivity is always true when <span>\\({\\mathbb {W}}_1\\boxtimes {\\mathbb {W}}_2\\)</span> is an irreducible and unitarizable <span>\\({\\mathbb {V}}\\)</span>-module. We give several applications: (1) We show that if <span>\\({\\mathbb {V}}\\)</span> is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group <i>G</i>, and if <span>\\({\\mathbb {V}}^G\\)</span> is strongly unitary, then <span>\\({\\mathbb {V}}^G\\)</span> is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if <span>\\({\\mathbb {V}}\\)</span> is unitary and strongly rational, and if <span>\\({\\mathbb {U}}\\)</span> is a simple current extension which is unitarizable as a <span>\\({\\mathbb {V}}\\)</span>-module, then <span>\\({\\mathbb {U}}\\)</span> is a unitary VOA.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-03-04","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/s00220-024-04959-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
A unitary and strongly rational vertex operator algebra (VOA) \({\mathbb {V}}\) is called strongly unitary if all irreducible \({\mathbb {V}}\)-modules are unitarizable. A strongly unitary VOA \({\mathbb {V}}\) is called completely unitary if for each unitary \({\mathbb {V}}\)-modules \({\mathbb {W}}_1,{\mathbb {W}}_2\) the canonical non-degenerate Hermitian form on the fusion product \({\mathbb {W}}_1\boxtimes {\mathbb {W}}_2\) is positive. It is known that if \({\mathbb {V}}\) is completely unitary, then the modular category \(\textrm{Mod}^\textrm{u}({\mathbb {V}})\) of unitary \({\mathbb {V}}\)-modules is unitary (Gui in Commun Math Phys 372(3):893–950, 2019), and all simple VOA extensions of \({\mathbb {V}}\) are automatically unitary and moreover completely unitary (Gui in Int Math Res Not 2022(10):7550–7614, 2022; Carpi et al. in Commun Math Phys 1–44, 2023). In this paper, we give a geometric characterization of the positivity of the Hermitian product on \({\mathbb {W}}_1\boxtimes {\mathbb {W}}_2\), which helps us prove that the positivity is always true when \({\mathbb {W}}_1\boxtimes {\mathbb {W}}_2\) is an irreducible and unitarizable \({\mathbb {V}}\)-module. We give several applications: (1) We show that if \({\mathbb {V}}\) is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group G, and if \({\mathbb {V}}^G\) is strongly unitary, then \({\mathbb {V}}^G\) is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if \({\mathbb {V}}\) is unitary and strongly rational, and if \({\mathbb {U}}\) is a simple current extension which is unitarizable as a \({\mathbb {V}}\)-module, then \({\mathbb {U}}\) is a unitary VOA.
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.