Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as w...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149140 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions / D.K. Wise // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 40 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862622963072761856 |
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| author | Wise, D.K. |
| author_facet | Wise, D.K. |
| citation_txt | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions / D.K. Wise // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 40 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.
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| first_indexed | 2025-11-30T03:15:39Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149140 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T03:15:39Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wise, D.K. 2019-02-19T17:38:16Z 2019-02-19T17:38:16Z 2009 Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions / D.K. Wise // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 40 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 22E70; 51P05; 53C80; 83C80; 83C99 https://nasplib.isofts.kiev.ua/handle/123456789/149140 Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'. This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. I thank James Dolan for many helpful discussions about geometry. I am also grateful for helpful discussions with John Baez, Steve Carlip, Stanley Deser, Stef fen Gielen, Andrew Waldron, and Joshua Willis. This work was supported in part by the National Science Foundation under grant DMS-0636297. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions Article published earlier |
| spellingShingle | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions Wise, D.K. |
| title | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions |
| title_full | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions |
| title_fullStr | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions |
| title_full_unstemmed | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions |
| title_short | Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions |
| title_sort | symmetric space cartan connections and gravity in three and four dimensions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149140 |
| work_keys_str_mv | AT wisedk symmetricspacecartanconnectionsandgravityinthreeandfourdimensions |