Killing (Super)Algebras Associated to Connections on Spinors
We generalise the notion of a Killing superalgebra, which arises in the physics literature on supergravity, to general dimension, signature, and choice of spinor module and Dirac current. We also allow for Lie algebras as well as superalgebras, capturing a set of examples previously defined using ge...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/214204 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Killing (Super)Algebras Associated to Connections on Spinors. Andrew D.K. Beckett. SIGMA 21 (2025), 081, 61 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862632475925151744 |
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| author | Beckett, Andrew D.K. |
| author_facet | Beckett, Andrew D.K. |
| citation_txt | Killing (Super)Algebras Associated to Connections on Spinors. Andrew D.K. Beckett. SIGMA 21 (2025), 081, 61 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We generalise the notion of a Killing superalgebra, which arises in the physics literature on supergravity, to general dimension, signature, and choice of spinor module and Dirac current. We also allow for Lie algebras as well as superalgebras, capturing a set of examples previously defined using geometric Killing spinors on higher-dimensional spheres. Our definition requires a connection on a spinor bundle - provided by supersymmetry transformations in the supergravity examples and by the Killing spinor equation on the spheres - and we obtain a set of sufficient conditions on such a connection for the Killing (super)algebra to exist. We show that these Lie (super)algebras are filtered deformations of graded subalgebras of (a generalisation of) the Poincaré superalgebra and then study such deformations abstractly using Spencer cohomology. In the highly supersymmetric Lorentzian case, we describe the filtered subdeformations which are of the appropriate form to arise as Killing superalgebras, lay out a classification scheme for their odd-generated subalgebras, and prove that, under certain technical conditions, there exist homogeneous Lorentzian spin manifolds on which these deformations are realised as Killing superalgebras. Our results generalise previous work in the 11-dimensional supergravity literature.
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| first_indexed | 2026-03-14T20:11:23Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-214204 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T20:11:23Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Beckett, Andrew D.K. 2026-02-20T08:04:43Z 2025 Killing (Super)Algebras Associated to Connections on Spinors. Andrew D.K. Beckett. SIGMA 21 (2025), 081, 61 pages 1815-0659 2020 Mathematics Subject Classification: 17B66; 83E50; 17B56 arXiv:2409.11306 https://nasplib.isofts.kiev.ua/handle/123456789/214204 https://doi.org/10.3842/SIGMA.2025.081 We generalise the notion of a Killing superalgebra, which arises in the physics literature on supergravity, to general dimension, signature, and choice of spinor module and Dirac current. We also allow for Lie algebras as well as superalgebras, capturing a set of examples previously defined using geometric Killing spinors on higher-dimensional spheres. Our definition requires a connection on a spinor bundle - provided by supersymmetry transformations in the supergravity examples and by the Killing spinor equation on the spheres - and we obtain a set of sufficient conditions on such a connection for the Killing (super)algebra to exist. We show that these Lie (super)algebras are filtered deformations of graded subalgebras of (a generalisation of) the Poincaré superalgebra and then study such deformations abstractly using Spencer cohomology. In the highly supersymmetric Lorentzian case, we describe the filtered subdeformations which are of the appropriate form to arise as Killing superalgebras, lay out a classification scheme for their odd-generated subalgebras, and prove that, under certain technical conditions, there exist homogeneous Lorentzian spin manifolds on which these deformations are realised as Killing superalgebras. Our results generalise previous work in the 11-dimensional supergravity literature. The author would like to thank José Figueroa-O’Farrill, under whose supervision this work was done, for his guidance and patience. Thanks also to Andrea Santi, James Lucietti, C.S. Shahbazi, and the sorely missed Paul de Medeiros for many enlightening conversations and helpful comments. Finally, thanks to the anonymous referees, whose comments and suggestions greatly improved the quality and clarity of this work. This work was carried out with scholarship funding from the Science and Technology Facilities Council (STFC) and the School of Mathematics at the University of Edinburgh. This work previously appeared as part of the author’s Ph.D. Thesis [8] but has not been published elsewhere. Some changes in notation and terminology have been made for this version to aid comparison with other work, and there are also some minor changes in exposition and corrections; in particular, Lemma 3.3 and its proof have been updated. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Killing (Super)Algebras Associated to Connections on Spinors Article published earlier |
| spellingShingle | Killing (Super)Algebras Associated to Connections on Spinors Beckett, Andrew D.K. |
| title | Killing (Super)Algebras Associated to Connections on Spinors |
| title_full | Killing (Super)Algebras Associated to Connections on Spinors |
| title_fullStr | Killing (Super)Algebras Associated to Connections on Spinors |
| title_full_unstemmed | Killing (Super)Algebras Associated to Connections on Spinors |
| title_short | Killing (Super)Algebras Associated to Connections on Spinors |
| title_sort | killing (super)algebras associated to connections on spinors |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214204 |
| work_keys_str_mv | AT beckettandrewdk killingsuperalgebrasassociatedtoconnectionsonspinors |