Generalised Umbral Moonshine
Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon, which connects finite groups...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210061 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Generalised Umbral Moonshine / M.C.N. Cheng, P. de Lange, D.P.Z. Whalen // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 57 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862607440943513600 |
|---|---|
| author | Cheng, M.C.N. de Lange, P. Whalen, D.P.Z. |
| author_facet | Cheng, M.C.N. de Lange, P. Whalen, D.P.Z. |
| citation_txt | Generalised Umbral Moonshine / M.C.N. Cheng, P. de Lange, D.P.Z. Whalen // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 57 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon, which connects finite groups and distinguished modular objects. In this paper, we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data for it. A central role is played by the deformed Drinfeld (or quantum) double of each umbral finite group G, specified by a cohomology class in H³(G, U(1)). We conjecture that in each of the 23 cases, there exists a rule to assign an infinite-dimensional module for the deformed Drinfeld double of the umbral finite group underlying the mock modular forms of umbral moonshine and generalised umbral moonshine. We also discuss the possible origin of the generalised umbral moonshine.
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| first_indexed | 2025-12-07T21:24:14Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210061 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:14Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cheng, M.C.N. de Lange, P. Whalen, D.P.Z. 2025-12-02T09:33:08Z 2019 Generalised Umbral Moonshine / M.C.N. Cheng, P. de Lange, D.P.Z. Whalen // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 57 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11F22; 11F37; 20C34 arXiv: 1608.07835 https://nasplib.isofts.kiev.ua/handle/123456789/210061 https://doi.org/10.3842/SIGMA.2019.014 Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon, which connects finite groups and distinguished modular objects. In this paper, we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data for it. A central role is played by the deformed Drinfeld (or quantum) double of each umbral finite group G, specified by a cohomology class in H³(G, U(1)). We conjecture that in each of the 23 cases, there exists a rule to assign an infinite-dimensional module for the deformed Drinfeld double of the umbral finite group underlying the mock modular forms of umbral moonshine and generalised umbral moonshine. We also discuss the possible origin of the generalised umbral moonshine. We are grateful to John Duncan, Simon Lentner, Terry Gannon, and Erik Verlinde for helpful discussions. We especially thank John Duncan for many of the group descriptions and for helpful comments on an earlier version of the manuscript. The work of M.C. and D.W. was supported by an ERC starting grant H2020 ERC StG 2014. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalised Umbral Moonshine Article published earlier |
| spellingShingle | Generalised Umbral Moonshine Cheng, M.C.N. de Lange, P. Whalen, D.P.Z. |
| title | Generalised Umbral Moonshine |
| title_full | Generalised Umbral Moonshine |
| title_fullStr | Generalised Umbral Moonshine |
| title_full_unstemmed | Generalised Umbral Moonshine |
| title_short | Generalised Umbral Moonshine |
| title_sort | generalised umbral moonshine |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210061 |
| work_keys_str_mv | AT chengmcn generalisedumbralmoonshine AT delangep generalisedumbralmoonshine AT whalendpz generalisedumbralmoonshine |