Vertex operator algebras, Higgs branches, and modular differential equations
Abstract Every four-dimensional N=2 $$ \mathcal{N}=2 $$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be co...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-08-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/JHEP08(2018)114 |
| _version_ | 1818294650357678080 |
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| author | Christopher Beem Leonardo Rastelli |
| author_facet | Christopher Beem Leonardo Rastelli |
| author_sort | Christopher Beem |
| collection | DOAJ |
| description | Abstract Every four-dimensional N=2 $$ \mathcal{N}=2 $$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any N=2 $$ \mathcal{N}=2 $$ SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class S $$ \mathcal{S} $$ theories, and N=2 $$ \mathcal{N}=2 $$ super Yang-Mills with sun $$ \mathfrak{s}\mathfrak{u}(n) $$ gauge group for small-to-moderate values of n. |
| first_indexed | 2024-12-13T03:35:07Z |
| format | Article |
| id | doaj.art-0b921ea7ca0a4559923ce48f7b314fcd |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-13T03:35:07Z |
| publishDate | 2018-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-0b921ea7ca0a4559923ce48f7b314fcd2022-12-22T00:01:04ZengSpringerOpenJournal of High Energy Physics1029-84792018-08-012018817210.1007/JHEP08(2018)114Vertex operator algebras, Higgs branches, and modular differential equationsChristopher Beem0Leonardo Rastelli1Mathematical Institute, University of OxfordC.N. Yang Institute for Theoretical Physics, Stony Brook UniversityAbstract Every four-dimensional N=2 $$ \mathcal{N}=2 $$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any N=2 $$ \mathcal{N}=2 $$ SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class S $$ \mathcal{S} $$ theories, and N=2 $$ \mathcal{N}=2 $$ super Yang-Mills with sun $$ \mathfrak{s}\mathfrak{u}(n) $$ gauge group for small-to-moderate values of n.http://link.springer.com/article/10.1007/JHEP08(2018)114Conformal and W SymmetryConformal Field TheoryExtended SupersymmetrySupersymmetric Gauge Theory |
| spellingShingle | Christopher Beem Leonardo Rastelli Vertex operator algebras, Higgs branches, and modular differential equations Journal of High Energy Physics Conformal and W Symmetry Conformal Field Theory Extended Supersymmetry Supersymmetric Gauge Theory |
| title | Vertex operator algebras, Higgs branches, and modular differential equations |
| title_full | Vertex operator algebras, Higgs branches, and modular differential equations |
| title_fullStr | Vertex operator algebras, Higgs branches, and modular differential equations |
| title_full_unstemmed | Vertex operator algebras, Higgs branches, and modular differential equations |
| title_short | Vertex operator algebras, Higgs branches, and modular differential equations |
| title_sort | vertex operator algebras higgs branches and modular differential equations |
| topic | Conformal and W Symmetry Conformal Field Theory Extended Supersymmetry Supersymmetric Gauge Theory |
| url | http://link.springer.com/article/10.1007/JHEP08(2018)114 |
| work_keys_str_mv | AT christopherbeem vertexoperatoralgebrashiggsbranchesandmodulardifferentialequations AT leonardorastelli vertexoperatoralgebrashiggsbranchesandmodulardifferentialequations |