Boundary algebras and Kac modules for logarithmic minimal models
Virasoro Kac modules were originally introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley–Lieb algebras. The lattice transfer operators include seams on the boundary that...
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| Format: | Article |
| Language: | English |
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Elsevier
2015-10-01
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| Series: | Nuclear Physics B |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S0550321315003028 |
| _version_ | 1818355619800809472 |
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| author | Alexi Morin-Duchesne Jørgen Rasmussen David Ridout |
| author_facet | Alexi Morin-Duchesne Jørgen Rasmussen David Ridout |
| author_sort | Alexi Morin-Duchesne |
| collection | DOAJ |
| description | Virasoro Kac modules were originally introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley–Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl–Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley–Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley–Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley–Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin–Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm–Gaberdiel–Kausch fusion algorithm. |
| first_indexed | 2024-12-13T19:44:12Z |
| format | Article |
| id | doaj.art-9cb4564cc39b45739a808e2f8fbaa040 |
| institution | Directory Open Access Journal |
| issn | 0550-3213 1873-1562 |
| language | English |
| last_indexed | 2024-12-13T19:44:12Z |
| publishDate | 2015-10-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Nuclear Physics B |
| spelling | doaj.art-9cb4564cc39b45739a808e2f8fbaa0402022-12-21T23:33:36ZengElsevierNuclear Physics B0550-32131873-15622015-10-01899C67776910.1016/j.nuclphysb.2015.08.017Boundary algebras and Kac modules for logarithmic minimal modelsAlexi Morin-Duchesne0Jørgen Rasmussen1David Ridout2Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Louvain-la-Neuve, B-1348, BelgiumSchool of Mathematics and Physics, University of Queensland, St Lucia, Brisbane, Queensland 4072, AustraliaDepartment of Theoretical Physics and Mathematical Sciences Institute, Australian National University, Acton, ACT 2601, AustraliaVirasoro Kac modules were originally introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley–Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl–Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley–Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley–Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley–Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin–Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm–Gaberdiel–Kausch fusion algorithm.http://www.sciencedirect.com/science/article/pii/S0550321315003028 |
| spellingShingle | Alexi Morin-Duchesne Jørgen Rasmussen David Ridout Boundary algebras and Kac modules for logarithmic minimal models Nuclear Physics B |
| title | Boundary algebras and Kac modules for logarithmic minimal models |
| title_full | Boundary algebras and Kac modules for logarithmic minimal models |
| title_fullStr | Boundary algebras and Kac modules for logarithmic minimal models |
| title_full_unstemmed | Boundary algebras and Kac modules for logarithmic minimal models |
| title_short | Boundary algebras and Kac modules for logarithmic minimal models |
| title_sort | boundary algebras and kac modules for logarithmic minimal models |
| url | http://www.sciencedirect.com/science/article/pii/S0550321315003028 |
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