| Title:
|
Cartan geometry, supergravity and group manifold approach (English) |
| Author:
|
François, Jordan |
| Author:
|
Ravera, Lucrezia |
| Language:
|
English |
| Journal:
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Archivum Mathematicum |
| ISSN:
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0044-8753 (print) |
| ISSN:
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1212-5059 (online) |
| Volume:
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60 |
| Issue:
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4 |
| Year:
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2024 |
| Pages:
|
243-281 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We make a case for the unique relevance of Cartan geometry for gauge theories of gravity and supergravity. We introduce our discussion by recapitulating historical threads, providing motivations. In a first part we review the geometry of classical gauge theory, as a background for understanding gauge theories of gravity in terms of Cartan geometry. The second part introduces the basics of the group manifold approach to supergravity, hinting at the deep rooted connections to Cartan supergeometry. The contribution is intended, not as an extensive review, but as a conceptual overview, and hopefully a bridge between communities in physics and mathematics. (English) |
| Keyword:
|
Cartan geometry |
| Keyword:
|
group manifold |
| Keyword:
|
classical gauge field theory of gravity |
| Keyword:
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Cartan supergeometry |
| Keyword:
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supergroup manifold |
| Keyword:
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supergravity |
| MSC:
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53Z05 |
| MSC:
|
58A32 |
| MSC:
|
58A50 |
| MSC:
|
83-01 |
| MSC:
|
83-03 |
| MSC:
|
83E50 |
| DOI:
|
10.5817/AM2024-4-243 |
| . |
| Date available:
|
2024-11-27T08:45:45Z |
| Last updated:
|
2024-11-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152643 |
| . |
| Reference:
|
[1] Abbott, L.F., Deser, S.: Charge definition in non-Abelian gauge theories.Phys. Lett. B 114 (4) (1982), 259–263. MR 0675235, 10.1016/0370-2693(82)90338-0 |
| Reference:
|
[2] Alekseevsky, D.V., Michor, P.W.: Differential geometry of Cartan connections.Publ. Math. Debrecen 47 (3–4) (1995), 349–375. MR 1362298, 10.5486/PMD.1995.1616 |
| Reference:
|
[3] Alfonsi, L.: The puzzle of global double field theory: Open problems and the case for a higher Kaluza-Klein perspective.Fortsch. Phys. 69 (7) (2021), 2000102. MR 4282510, 10.1002/prop.202000102 |
| Reference:
|
[4] Andrianopoli, L., D’Auria, R.: N=1 and N=2 pure supergravities on a manifold with boundary.JHEP 08 (2014), 012. 10.1007/JHEP08(2014)012 |
| Reference:
|
[5] Andrianopoli, L., D’Auria, R., Ravera, L.: Hidden Gauge structure of supersymmetric free differential algebras.JHEP 08 (2016), 095. MR 3555659, 10.1007/JHEP08(2016)095 |
| Reference:
|
[6] Andrianopoli, L., D’Auria, R., Ravera, L.: More on the hidden symmetries of 11D supergravity.Phys. Lett. B 772 (2017), 578–585. 10.1016/j.physletb.2017.07.016 |
| Reference:
|
[7] Andrianopoli, L., Ravera, L.: On the geometric approach to the boundary problem in supergravity.Universe 7 (12) (2021), 463. 10.3390/universe7120463 |
| Reference:
|
[8] Attard, J., François, J.: Tractors and Twistors from conformal Cartan geometry: a gauge theoretic approach II. Twistors.Classical Quantum Gravity 34 (8) (2017), 28 pp. MR 3633535 |
| Reference:
|
[9] Attard, J., François, J.: Tractors and twistors from conformal Cartan geometry: a Gauge theoretic approach I. Tractors.Adv. Theor. Math. Phys. 22 (8) (2018), 1831–1883. MR 3984517, 10.4310/ATMP.2018.v22.n8.a1 |
| Reference:
|
[10] Attard, J., François, J., Lazzarini, S.: Weyl gravity and Cartan geometry.Phys. Rev. D 93 (2016), 085032. MR 3550958, 10.1103/PhysRevD.93.085032 |
| Reference:
|
[11] Attard, J., François, J., Lazzarini, S., Masson, T.: Cartan connections and Atiyah Lie algebroids.J. Geom. Phys. 148 (2020), 103541. MR 4034711, 10.1016/j.geomphys.2019.103541 |
| Reference:
|
[12] Baez, J.C., Huerta, J.: An invitation to higher Gauge theory.Gen. Relativity Gravitation 43 (9) (2011), 2335–2392. MR 2825807, 10.1007/s10714-010-1070-9 |
| Reference:
|
[13] Bailey, T.N., Eastwood, M.G., Gover, A.R.: Thomas’s structure bundle for conformal, projective and related structures.Rocky Mountain J. Math. 24 (4) (1994), 1191–1217. Zbl 0828.53012, MR 1322223, 10.1216/rmjm/1181072333 |
| Reference:
|
[14] Becchi, C., Rouet, A., Stora, R.: Renormalization of the Abelian Higgs-Kibble Model.Comm. Math. Phys. 42 (1975), 127–162. MR 0389060, 10.1007/BF01614158 |
| Reference:
|
[15] Becchi, C., Rouet, A., Stora, R.: Renormalization of Gauge theories.Ann. Physics 98 (1976), 287–321. MR 0413861, 10.1016/0003-4916(76)90156-1 |
| Reference:
|
[16] Berezin, F.A.: The method of second quantization.Pure Appl. Phys., New York, Academic Press, 1966. MR 0208930 |
| Reference:
|
[17] Berezin, F.A.: Introduction to Superanalysis.Mathematical Physics and Applied Mathematics, Springer, 1st ed., 1987. MR 0914369 |
| Reference:
|
[18] Berezin, F.A., Kac, G.I.: Lie groups with commuting and anticommuting parameters.Sb. Math. 11 (3) (1970), 311–325. MR 0265520, 10.1070/SM1970v011n03ABEH001137 |
| Reference:
|
[19] Berezin, F.A., Marinov, M.S.: Particle spin dynamics as the grassmann variant of classical mechanics.Ann. Physics 104 (2) (1977), 336–362. 10.1016/0003-4916(77)90335-9 |
| Reference:
|
[20] Berghofer, P., François, J., Friederich, S., Gomes, H., Hetzroni, G., Maas, A., Sondenheimer, R.: Gauge Symmetries, Symmetry Breaking, and Gauge-Invariant Approaches.Elements in the Foundations of Contemporary Physics, Cambridge University Press, 2023. |
| Reference:
|
[21] Bertlmann, R.A.: Anomalies In Quantum Field Theory.International Series of Monographs on Physics, vol. 91, Oxford University Press, 1996., 1996. MR 1373240 |
| Reference:
|
[22] Blagojević, M., Hehl, F.W., Kibble, T.W.B.: Gauge Theories of Gravitation.Imperial College Press, 2013. |
| Reference:
|
[23] Blagojević, M., Hehland, F.W., Kibble, T.W.B.: Gauge Theories of Gravitation.Imperial College Press, 2013. |
| Reference:
|
[24] Bonor, L.: Fermions and Anomalies in Quantum Field Theories.Theoretical and Mathematical Physics, Springer Cham, 1st ed., 2023. MR 4590535 |
| Reference:
|
[25] Bonora, L., Cotta-Ramusino, P.: Some remark on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations.Comm. Math. Phys. 87 (1983), 589–603. MR 0691046, 10.1007/BF01208267 |
| Reference:
|
[26] Burdet, G., Duval, C., Perrin, M.: Cartan structures on galilean manifolds: The chronoprojective geometry.J. Math. Phys. 24 (7) (1983), 1752–1760. MR 0709508, 10.1063/1.525927 |
| Reference:
|
[27] Burdet, G., Duval, C., Perrin, M.: Chronoprojective Cartan Structures on four-dimensional manifolds.RIMS, Kyoto Univ. 19 (1983), 813–840. MR 0716977, 10.2977/prims/1195182453 |
| Reference:
|
[28] Čap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory.Mathematical Surveys and Monographs, vol. 1, American Mathematical Society, 2009. Zbl 1183.53002, MR 2532439, 10.1090/surv/154 |
| Reference:
|
[29] Čap, A., Slovák, J., Souček, V.: Invariant operators on manifolds with almost hermitian symmetric structures. I. Invariant differentiation.Acta Math. Univ. Comenian. 66 (1997), 33–69. MR 1474550 |
| Reference:
|
[30] Čap, A., Slovák, J., Souček, V.: Invariant operators on manifolds with almost hermitian symmetric structures. II. Normal Cartan connections.Acta Math. Univ. Comenian. 66 (1997), 203–220. MR 1620484 |
| Reference:
|
[31] Čap, A., Slovák, J., Souček, V.: Invariant operators on manifolds with almost hermitian symmetric structures,.III. Standard operators.Di erential Geom. Appl. 12 (1) (2000), 51–84. MR 1757020, 10.1016/S0926-2245(00)00003-6 |
| Reference:
|
[32] Cartan, É.: Les espaces à connexion conforme.Ann. Polon. Math. 2 (1923), 171–221. |
| Reference:
|
[33] Cartan, É.: Les récentes généralisations de la notion d’espace.Bull. Sci. Math. 48 (1924), 825–861. |
| Reference:
|
[34] Cartan, É.: Sur les variétés à connexion projective.Bull. Soc. Math. France 52 (1924), 205–241. MR 1504846, 10.24033/bsmf.1053 |
| Reference:
|
[35] Castellani, L.: A geometric interpretation of BRST symmetry.Classical Quantum Gravity 7 (1990), L159–164. MR 1064172, 10.1088/0264-9381/7/8/001 |
| Reference:
|
[36] Castellani, L., Catenacci, R., Grassi, P.A.: Supergravity actions with integral forms.Nuclear Phys. B 889 (2014), 419–442. MR 3280375, 10.1016/j.nuclphysb.2014.10.023 |
| Reference:
|
[37] Castellani, L., Catenacci, R., Grassi, P.A.: The geometry of supermanifolds and new supersymmetric actions.Nuclear Phys. B 899 (2015), 112–148. MR 3398909 |
| Reference:
|
[38] : Tullio Regge: An Eclectic Genius: From Quantum Gravity to Computer Play.vol. 9, World Scientific, 2019. |
| Reference:
|
[39] Castellani, L., D’Auria, R., Fré, P.: Supergravity and superstrings: A Geometric perspective.vol. 1–3, Singapore: World Scientific, 1991. MR 1120023 |
| Reference:
|
[40] Castellani, L., D’Auria, R., Fré, P.: Supergravity and superstrings: A Geometric perspective. Vol. 2: Supergravity.World Scientific Pub. Co. Inc., 1991. MR 1120023 |
| Reference:
|
[41] Castellani, L., Grassi, P.A.: Hodge duality and supergravity.Phys. Rev. D 108 (4) (2023), 046018. MR 4646295, 10.1103/PhysRevD.108.046018 |
| Reference:
|
[42] Catenacci, R., Debernardi, M., Grassi, P.A., Matessi, D.: Cech and de Rham cohomology of integral forms.J. Geom. Phys. 62 (2012), 890–902. MR 2888990, 10.1016/j.geomphys.2011.12.011 |
| Reference:
|
[43] Catenacci, R., Pirola, G.P.: A geometrical description of local and global anomalies.Lett. Math. Phys. 19 (1990), 45–51. MR 1040409, 10.1007/BF00402259 |
| Reference:
|
[44] Catenacci, R., Pirola, G.P., Martellini, M., Reina, C.: Group actions and anomalies in gauge theories.Phys. Lett. B 172 (1986), 223–226, https://doi.org/10.1016/0370-2693(86)90839-7. MR 0844737, 10.1016/0370-2693(86)90839-7 |
| Reference:
|
[45] Chamseddine, A.H., West, P.C.: Supergravity as a Gauge theory of supersymmetry.Nuclear Phys. 129 (1977), 39–44. 10.1016/0550-3213(77)90018-9 |
| Reference:
|
[46] Choquet-Bruhat, Y.: General Relativity and the Einstein Equations.Oxford Mathematical Monographs, Oxford University Press, 2009. MR 2473363 |
| Reference:
|
[47] Coleman, S., Mandula, J.: All possible symmetries of the S matrix.Phys. Rev. 159 (1967), 1251–1256. 10.1103/PhysRev.159.1251 |
| Reference:
|
[48] Concha, P., Ravera, L., Rodríguez, E.: On the supersymmetry invariance of flat supergravity with boundary.JHEP 01 (2018), 192. MR 3919303 |
| Reference:
|
[49] Connes, A.: Noncommutative Geometry Year 2000.Birkhäuser Basel, 2000, 481–559. MR 1826266 |
| Reference:
|
[50] Connes, A., Marcolli, M.: An Invitation to Noncommutative Geometry.ch. A walk in the non-commutative garden, pp. 1–128, World Scientific Publishing Company, 2008. MR 2408150 |
| Reference:
|
[51] Cotta Ramusino, P., Reina, C.: The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories.J. Geom. Phys. 1 (3) (1984), 121–155. MR 0828400, 10.1016/0393-0440(84)90022-6 |
| Reference:
|
[52] Cremmer, E., Julia, B., Scherk, J.: Supergravity theory in 11 dimensions.Phys. Lett. B 76 (4) (1978), 409–412. MR 0550003, 10.1016/0370-2693(78)90894-8 |
| Reference:
|
[53] Cremonini, C.A.: The geometry of picture changing operators.accepted for publication in Adv. Theor. Math. Phys.; arXiv:2305.02828 [math-ph]. MR 4806830 |
| Reference:
|
[54] Cremonini, C.A., Grassi, P.A.: Generalised cocycles and super p-branes.6 2022. MR 4761204 |
| Reference:
|
[55] Cremonini, C.A., Grassi, P.A.: Pictures from super Chern-Simons theory.JHEP 03 (2020), 043. MR 4090094, 10.1007/JHEP03(2020)043 |
| Reference:
|
[56] Cremonini, C.A., Grassi, P.A.: Super Chern-Simons theory: Batalin-Vilkovisky formalism and $A_\infty $ algebras.Phys. Rev. D 102 (2) (2020), 025009. MR 4134650, 10.1103/PhysRevD.102.025009 |
| Reference:
|
[57] Cremonini, C.A., Grassi, P.A., Noris, R., Ravera, L.: Supergravities and branes from Hilbert-Poincaré series.JHEP 12 (2023), 088. MR 4685988, 10.1007/JHEP12(2023)088 |
| Reference:
|
[58] D’Auria, R.: Geometric supergravity.5 2020. |
| Reference:
|
[59] D’Auria, R., Fré, P.: Cartan integrable systems, that is differential free algebras, in supergravity.September School on Supergravity and Supersymmetry, vol. 9, 1982. MR 0728778 |
| Reference:
|
[60] D’Auria, R., Fré, P.: Geometric supergravity in d = 11 and its hidden supergroup.Nuclear Phys. B 201 (1982), 101–140, [Erratum: Nucl.Phys.B 206, 496 (1982)]. MR 0667482 |
| Reference:
|
[61] D’Auria, R., Ravera, L.: Conformal gravity with totally antisymmetric torsion.Phys. Rev. D 104 (8) (2021), 084034. MR 4341563, 10.1103/PhysRevD.104.084034 |
| Reference:
|
[62] : Quantum fields and strings: A course for mathematicians. Vol. 1, 2.Providence, RI: AMS, American Mathematical Society, 1999. |
| Reference:
|
[63] DeWitt, B.: Supermanifolds.Cambridge University Press, 1984. Zbl 0551.53002, MR 0778559 |
| Reference:
|
[64] Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry and new models of Gauge theory.J. Math. Phys. 31 (2) (1990), 323–330. MR 1034168, 10.1063/1.528917 |
| Reference:
|
[65] Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry of matrix algebras.J. Math. Phys. 31 (2) (1990), 316–322. Zbl 0704.53081, MR 1034167, 10.1063/1.528916 |
| Reference:
|
[66] Eastwood, M., Bailey, T.: Complex paraconformal manifolds – their differential geometry and twistor theory.Forum Math. 3 (1) (1991), 61–103. MR 1085595 |
| Reference:
|
[67] Eder, K.: Super Cartan geometry and the super Ashtekar connection.Ann. Henri Poincaré 24 (2023), 3531–3599. MR 4642269, 10.1007/s00023-023-01290-5 |
| Reference:
|
[68] Eder, K., Huerta, J., Noja, S.: Poincaré Duality for Supermanifolds, Higher Cartan Structures and Geometric Supergravity.12 2023. |
| Reference:
|
[69] Egeileh, M., El Chami, F.: Some remarks on the geometry of superspace supergravity.J. Geom. Phys. 62 (1) (2012), 53–60. MR 2854194, 10.1016/j.geomphys.2011.09.008 |
| Reference:
|
[70] Eguchi, T., Gilkey, P., Hanson, A.: Gravitation, gauge theories and differential geometry.Phys. Rep. 66 (6) (1980), 213–393. MR 0598586, 10.1016/0370-1573(80)90130-1 |
| Reference:
|
[71] Ehresmann, C.: Sur la théorie des espaces fibrés.Colloque de topologie algébrique du CNRS, Paris, 1947, pp. 3–15. MR 0035021 |
| Reference:
|
[72] Ehresmann, C., Collectif, : Les connexions infinitésimales dans un espace fibré différentiable.Séminaire Bourbaki : années 1948/49 – 1949/50 – 1950/51, exposés 1–49, no. 1, Société mathématique de France, 1952, pp. 153–168. MR 0042768 |
| Reference:
|
[73] Faddeev, L.D., Popov, V.N.: Feynman diagrams for the Yang-Mills field.Phys. Lett. B 25 (1) (1967), 29–30. 10.1016/0370-2693(67)90067-6 |
| Reference:
|
[74] Ferrara, S., Kaku, M., Townsend, P.K., van Nieuwenhuizen, P.: Gauging the graded conformal group with unitary internal symmetries.Nuclear Pkhys. B129 (1977), 125. |
| Reference:
|
[75] Fine, D., Fine, A.: Gauge theory, anomalies and global geometry: The interplay of physics and mathematics.Stud. Hist. Philos. Modern Phys. 28 (1997), 307–323. MR 1604088, 10.1016/S1355-2198(97)00011-7 |
| Reference:
|
[76] Fiorenza, D., Sati, H., Schreiber, U.: Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields.Int. J. Geom. Methods Mod. Phys. 12 (2014), 1550018. MR 3305054, 10.1142/S0219887815500188 |
| Reference:
|
[77] Fiorenza, D., Sati, H., Schreiber, U.: The Rational higher structure of M-theory.Fortsch. Phys. 67 (8–9) (2019*), 1910017. MR 4016023, 10.1002/prop.201910017 |
| Reference:
|
[78] Fiorenza, D., Sati, H., Schreiber, U.: Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5.JHEP 02 (2020), 107. MR 4089174, 10.1007/JHEP02(2020)107 |
| Reference:
|
[79] François André, J.: The dressing field method for diffeomorphisms: a relational framework.2023. MR 4771773 |
| Reference:
|
[80] François, J.: Artificial versus substantial Gauge symmetries: A criterion and an application to the electroweak model.Philos. Sci. 86 (3) (2019), 472–496. MR 3977906, 10.1086/703571 |
| Reference:
|
[81] François, J.: Dilaton from tractor and matter field from twistor.J. High Energy Phys. 2019 (6) (2019), 18. MR 3982522, 10.1007/JHEP06(2019)018 |
| Reference:
|
[82] François, J.: Bundle geometry of the connection space, covariant hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method.J. High Energy Phys. 2021 (3) (2021), 113 p. MR 4261003 |
| Reference:
|
[83] François, J.: Differential geometry of Gauge theory: An introduction.PoS, Modave 2020 (2021), 002. |
| Reference:
|
[84] François, J., Parrini, N., Boulanger, N.: Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions.J. High Energy Phys. 2021 (12) (2021), 186. MR 4369415 |
| Reference:
|
[85] Friedrich, H.: Twistor connection and normal conformal Cartan connection.Gen. Relativity Gravitation 8 (5) (1977), 303–312. MR 0474086, 10.1007/BF00771141 |
| Reference:
|
[86] Galperin, A.S., Ivanov, E.A., Ogievetsky, V.I., Sokatchev, E.S.: Harmonic superspace.Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2007. MR 2450736 |
| Reference:
|
[87] Giovanelli, M.: Nothing but coincidences: the point-coincidence and einstein’s struggle with the meaning of coordinates in physics.European J. Philos. Sci. 11 (2) (2021), 45. MR 4254163, 10.1007/s13194-020-00332-7 |
| Reference:
|
[88] Golfand, Y.A., Likhtman, E.P.: Extension of the algebra of Poincare group generators and violation of p invariance.JETP Lett. 13 (1971), 323–326. MR 0672786 |
| Reference:
|
[89] Gover, R.: Aspects of parabolic invariant theory.Proceedings of the 18th Winter School “Geometry and Physics”, 1999, pp. 25–47. MR 1692257 |
| Reference:
|
[90] Hamilton, M.: Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics.1st ed., Universitext. Springer, 2018. MR 3837560 |
| Reference:
|
[91] Hélein, F.: Gauge and gravity theories on a dynamical principal bundle.10 2023. |
| Reference:
|
[92] Hélein, F., Vey, D.: Curved space-times by crystallization of liquid fiber bundles.Found. Phys. 47 (1) (2017), 1–41. MR 3599605, 10.1007/s10701-016-0039-2 |
| Reference:
|
[93] Hersent, K., Mathieu, P., Wallet, J.C.: Gauge theories on quantum spaces.Phys. Rep. 1014 (2023), 1–83. MR 4567524, 10.1016/j.physrep.2023.03.002 |
| Reference:
|
[94] Jurčo, B., Sämann, C., Schreiber, U., Wolf, M.: Higher sructures in M-theory.Fortsch. Phys. 67 (8–9) (2019), 1910004. MR 4016007 |
| Reference:
|
[95] Jurčo, B., Sämann, C., Wolf, M.: Semistrict higher Gauge theory.JHEP 2015 (4) (2015), 1–67. MR 3351282 |
| Reference:
|
[96] Jurčo, B., Sämann, C., Wolf, M.: Higher groupoid bundles, higher spaces, and self-dual tensor field equations.Fortsch. Phys. 64 (8–9 (2016), 674–717. MR 3548195, 10.1002/prop.201600031 |
| Reference:
|
[97] Kaku, M., Townsend, P.K., van Nieuwenhuizen, P.: Gauge theory of the conformal and superconformal group.Phys. lett. 69B (1977), 304–308. MR 0446293, 10.1016/0370-2693(77)90552-4 |
| Reference:
|
[98] Kaku, M., Townsend, P.K., van Nieuwenhuizen, P.: Properties of conformal supergravity.Phys. Rev. D17 (1978), 3179. MR 0523881 |
| Reference:
|
[99] Kasprzak, P.: On a certain approach to quantum homogeneous spaces.Comm. Math. Phys. 313 (1) (2012), 237–255. MR 2928224, 10.1007/s00220-012-1491-2 |
| Reference:
|
[100] Kastor, D.: Komar integrals in higher (and lower) derivative gravity.Classical Quantum Gravity 25 (17) (2008), 175007. MR 2430676, 10.1088/0264-9381/25/17/175007 |
| Reference:
|
[101] Kibble, T.W.B.: Lorentz invariance and the gravitational field.J. Math. Phys. 2 (2) (1961), 212–221. MR 0127952, 10.1063/1.1703702 |
| Reference:
|
[102] Kim, H., Saemann, C.: Adjusted parallel transport for higher Gauge theories.J. Phys. A 53 (44) (2020), 445206. MR 4177087, 10.1088/1751-8121/ab8ef2 |
| Reference:
|
[103] Kobayashi, S.: On connections of Cartan.Canad. J. Math. 8 (1956), 145–156. MR 0077978, 10.4153/CJM-1956-018-8 |
| Reference:
|
[104] Kobayashi, S.: Theory of connections.Ann. Mat. Pura Appl. 43 (1) (1957), 119–194. MR 0096276, 10.1007/BF02411907 |
| Reference:
|
[105] Kobayashi, S.: Canonical forms on frame bundles of higher order contact.Proc. Sympos. Pure Math. American Mathematical Society III (1961), 186–193. MR 0126810 |
| Reference:
|
[106] Kobayashi, S.: Transformation Groups in Differential Geometry.Springer, 1972. Zbl 0246.53031, MR 0355886 |
| Reference:
|
[107] Kobayashi, S., Nagano, T.: On projective connections.J. Math. Mech. 13 (1964), 215–235. MR 0159284 |
| Reference:
|
[108] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry.vol. I, Wiley & Sons, 1963. Zbl 0119.37502, MR 0152974 |
| Reference:
|
[109] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry.vol. II, Wiley & Sons, 1969. Zbl 0175.48504, MR 0238225 |
| Reference:
|
[110] Kolar, I., Michor, P., Slovak, J.: Natural Operations in Differential Geometry.Springer-Verlag Berlin, 1993. MR 1202431 |
| Reference:
|
[111] Korzyński, M., Lewandowski, J.: The normal conformal Cartan connection and the Bach tensor.Classical Quantum Gravity 20 (16) (2003), 3745–3764. MR 2001693, 10.1088/0264-9381/20/16/314 |
| Reference:
|
[112] Kostant, B.: Graded manifolds, graded Lie theory, and prequantization.Differ. geom. Meth. math. Phys., Proc. Symp. Bonn 1975, Lect. Notes Math., 1977. MR 0580292 |
| Reference:
|
[113] Lazzarini, S., Tidei, C.: Polyakov soldering and second-order frames: The role of the cartan connection.Lett. Math. Phys. 85 (1) (2008), 27–37. MR 2425659, 10.1007/s11005-008-0253-8 |
| Reference:
|
[114] Leites, D.: Introduction to the theory of supermanifolds.Russian Math. Surveys 35 (1) (1980), 1–64. Zbl 0462.58002, MR 0565567, 10.1070/RM1980v035n01ABEH001545 |
| Reference:
|
[115] Marle, C.M.: The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles, in Geometry and Topology of Manifolds.vol. 76, Banach Center Publication, 2007. MR 2342856 |
| Reference:
|
[116] McDowell, S.W., Mansouri, F.: Unified geometric theory of gravity and supergravity.Phys. Rev. Lett. 38 (1977), 739–742. MR 0676657, 10.1103/PhysRevLett.38.739 |
| Reference:
|
[117] Merkulov, S.A.: A conformally invariant theory of gravitation and electromagnetism.Classical Quantum Gravity 1 (4) (1984), 349. MR 0758513, 10.1088/0264-9381/1/4/007 |
| Reference:
|
[118] Merkulov, S.A.: The twistor connection and Gauge invariance principle.Comm. Math. Phys. 93 (3) (1984), 325–331. MR 0745687, 10.1007/BF01258531 |
| Reference:
|
[119] Molotkov, V.: Infinite-dimensional and colored supermanifolds.J. Nonlinear Math. Phys. 17 (2021), 375–446. MR 2827484, 10.1142/S140292511000088X |
| Reference:
|
[120] Nakahara, M.: Geometry, Topology and Physics. 2nd Edition..Graduate Student Series in Physics, Taylor & Francis, 2003. MR 2001829 |
| Reference:
|
[121] Ne’eman, Y., Regge, T.: Gravity and spergravity as Gauge theories on a group manifold.Riv. Nuovo Cim 1 (5) (1978), 1–43. MR 0507169, 10.1007/BF02724472 |
| Reference:
|
[122] Ne’eman, Y., Regge, T.: Gravity and supergravity as Gauge theories on a group manifold.Phys. Lett. B 74 (1978), 54–56. 10.1016/0370-2693(78)90058-8 |
| Reference:
|
[123] Ó Buachalla, R.: Noncommutative complex structures on quantum homogeneous spaces.J. Geom. Phys. 99 (2016), 154–173. MR 3428362, 10.1016/j.geomphys.2015.10.003 |
| Reference:
|
[124] Ochiai, T.: Geometry associated with semi-simple homogeneous spaces.Transactions of the American Mathematical Society 152 (1970), 159–193. MR 0284936, 10.1090/S0002-9947-1970-0284936-6 |
| Reference:
|
[125] Ogiue, K.: Theory of conformal connections.Kodai Math. Sem. Rep. 19 (1967), 193–224. MR 0217723, 10.2996/kmj/1138845392 |
| Reference:
|
[126] O’Raifeartaigh, L.: The dawning of Gauge theory.Princeton Series in Physics, Princeton University Press, 1997. MR 1374603 |
| Reference:
|
[127] Penrose, R.: The twistor program.Rep. Mathematical Phys 12 (1977), 65–76. MR 0465032, 10.1016/0034-4877(77)90047-7 |
| Reference:
|
[128] Penrose, R., MacCallum, M.A.H.: Twistor theory: An approach to the quantisation of fields and space-time..Phys. Rep. 6 (4) (1973), 241–316. MR 0475660, 10.1016/0370-1573(73)90008-2 |
| Reference:
|
[129] Penrose, R., Rindler, W.: Spinors and Space-Time.vol. 1, Cambridge University Press, 1984. MR 0838301 |
| Reference:
|
[130] Penrose, R., Rindler, W.: Spinors and Space-Time.vol. 2, Cambridge University Press, 1986. MR 0838301 |
| Reference:
|
[131] Ravera, L.: On the hidden symmetries of D=11 supergravity.Springer Proc. Math. Stat. 396 (2022), 211–222. MR 4607963 |
| Reference:
|
[132] Ravera, L.: On the role of torsion and higher forms in off-shell supergravity.Fortsch. Phys. 71 (8–9) (2023), 2300036. MR 4637988, 10.1002/prop.202300036 |
| Reference:
|
[133] Rogers, A.: A global theory of supermanifolds.J. Math. Phys. 21 (6) (1980), 1352–1365. Zbl 0447.58003, MR 0574696, 10.1063/1.524585 |
| Reference:
|
[134] Rogers, A.: Supermanifolds.World Scientific Publishing Company, 2007. MR 2320438 |
| Reference:
|
[135] Rovelli, C: Partial observables.Phys. Rev. D 65 (2002), 124013. MR 1926193, 10.1103/PhysRevD.65.124013 |
| Reference:
|
[136] Rovelli, C.: Quantum gravity.Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2004. MR 2106565 |
| Reference:
|
[137] Rovelli, C.: Why gauge?.Found. Phys. 44 (2014), 91–104. |
| Reference:
|
[138] Rovelli, C., Vidotto, F.: Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory.Cambridge University Press, 2014. |
| Reference:
|
[139] Rüdiger, R., Penrose, R.: The Dirac equation and spinning particles in general relativity.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, vol. 377 (1771), 1981, pp. 417–424. MR 0629659 |
| Reference:
|
[140] Sachse, C.: A categorical formulation of superalgebra and supergeometry.arXiv:0802.4067, 2008. |
| Reference:
|
[141] Sachse, C., Wockel, C.: The diffeomorphism supergroup of a finite-dimensional supermanifold.Adv. Theor. Math. Phys. 15 (2) (2012), 285–323. MR 2924232, 10.4310/ATMP.2011.v15.n2.a2 |
| Reference:
|
[142] Salam, A., Strathdee, J.: Supersymmetry and superfields.Fortschr. Phys. 26 (2) (1978), 57–142. MR 0495883, 10.1002/prop.19780260202 |
| Reference:
|
[143] Sati, H., Schreiber, U., Stasheff, J.: $L_{\infty }$ algebra connections and applications to String- and Chern-Simons n-transport.2 2008. MR 2742762 |
| Reference:
|
[144] Sauer, T.: Field equations in teleparallel space-time: Einstein’s fernparallelismus approach toward unified field theory.Historia Math. 33 (4) (2006), 399–439, Special Issue on Geometry and its Uses in Physics, 1900-1930. MR 2276051, 10.1016/j.hm.2005.11.005 |
| Reference:
|
[145] Schmitt, T.: Supergeometry and quantum field theory, or: What is a classical configuration?.Rev. Math. Phys. 09(08) (1997), 993–1052. MR 1487885, 10.1142/S0129055X97000348 |
| Reference:
|
[146] Schneider, H.-J.: Principal homogeneous spaces for arbitrary hopf algebras.Israel J. Math. 72 (1) (1990), 167–195. MR 1098988, 10.1007/BF02764619 |
| Reference:
|
[147] Schreiber, U.: Differential cohomology in a cohesive $\infty $-topos.arXiv:1310.7930 [math-ph]. |
| Reference:
|
[148] Sciama, D.W.: The physical structure of general relativity.Rev. Modern Phys. 36 (1964), 463–469. 10.1103/RevModPhys.36.463 |
| Reference:
|
[149] Sharpe, R.W.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program.Graduate text in Mathematics, vol. 166, Springer, 1996. MR 1453120 |
| Reference:
|
[150] Shvarts, A.S.: On the definition of superspace.Theoret. and Math. Phys. 60 91) (1984), 657–660. Zbl 0575.58005, MR 0760438, 10.1007/BF01018248 |
| Reference:
|
[151] Singer, I.M.: Some remark on the Gribov ambiguity.Comm. Math. Phys. 60 (1978), 7–12. MR 0500248, 10.1007/BF01609471 |
| Reference:
|
[152] Singer, I.M.: The geometry of the orbit space for non-abelian gauge theories.Physica Scripta 24 (5) (1981), 817–820. MR 0639408, 10.1088/0031-8949/24/5/002 |
| Reference:
|
[153] Stachel, J.: The hole argument and some physical and philosophical implications.Living Reviews in Relativity 17 (1) (2014), 1. 10.12942/lrr-2014-1 |
| Reference:
|
[154] Steenrod, N.: The Topology of Fibre Bundles. (PMS–14).Princeton University Press, 1951. MR 0039258 |
| Reference:
|
[155] Stelle, K.S., West, P.C.: de Sitter gauge invariance and the geometry of the Einstein-Cartan theory.J. Phys. A. 12 (1979), 205–210. MR 0536913, 10.1088/0305-4470/12/8/003 |
| Reference:
|
[156] Stora, R.: Algebraic structure and topological origin of chiral anomalies.Progress in Gauge Field Theory, Cargese 1983, vol. 115, NATO ASI Ser.B,, Plenum Press, 1984. MR 0782343 |
| Reference:
|
[157] Sullivan, D.: Infinitesimal computations in topology.Publ. Math. Inst. Hautes Études Sci. 47 (1977), 269–331. MR 0646078, 10.1007/BF02684341 |
| Reference:
|
[158] Tamborino, J.: Relational observables in gravity: a review.SIGMA 8 (2012), 017. MR 2942822 |
| Reference:
|
[159] Tanaka, N.: Projective connections and projective transformations.Nagoya Math. J. 12 (1957), 1–24. MR 0105154, 10.1017/S0027763000021905 |
| Reference:
|
[160] Tyutin, L.V.: Gauge invariance in field theory and statistical physics in operator formalism.Lebedev preprint FIAN, 39:1975. |
| Reference:
|
[161] Unzicker, A., Case, T.: Translation of Einstein’s Attempt of a Unified Field Theory with Teleparallelism.arXiv:physics/0503046 [physics.hist-ph], 2005. |
| Reference:
|
[162] Utiyama, R.: Invariant Theoretical Interpretation of Interaction.Phys. Rev. 101 (1956), 1597–1607. MR 0078223, 10.1103/PhysRev.101.1597 |
| Reference:
|
[163] Volkov, D.V., Akulov, V.P.: Is the neutrino a goldstone particle?.Phys. Lett. B 46 (1) (1973), 109–110. 10.1016/0370-2693(73)90490-5 |
| Reference:
|
[164] Voronov, A.A.: Mappings of supermanifolds.Theoret. and Math. Phys. 60 (1) (1984), 660–664. MR 0760439, 10.1007/BF01018249 |
| Reference:
|
[165] Wess, J., Zumino, B.: Supergauge transformations in four-dimensions.Nuclear Phys. 70 (1974), 39–50. MR 0356830, 10.1016/0550-3213(74)90355-1 |
| Reference:
|
[166] Wess, J., Zumino, B.: Superspace formulation of supergravity.Phys. Lett. B 66 (4) (1977), 361–364. MR 0456294, 10.1016/0370-2693(77)90015-6 |
| Reference:
|
[167] Weyl, H.: Gravitation and electricity.vol. 1918, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1918. |
| Reference:
|
[168] Weyl., H.: A new extension of relativity theory.Ann. Physics 59 (1919), 101–133. |
| Reference:
|
[169] Weyl, H.: Gravitation and the electron.Proceedings of the National Academy of Sciences, vol. 15 (4), 1929, pp. 323–334. |
| Reference:
|
[170] Wise, D.K.: Topological Gauge Theory, Cartan Geometry, and Gravity.Ph.D. thesis, 2007. MR 2710391 |
| Reference:
|
[171] Wise, D.K.: Symmetric space, cartan connections and gravity in three and four dimensions.SIGMA 5 (2009), 080–098. MR 2529167 |
| Reference:
|
[172] Wise, D.K.: MacDowell–Mansouri gravity and Cartan geometry.Classical Quantum Gravity 27 (2010), 155010. MR 2659245, 10.1088/0264-9381/27/15/155010 |
| Reference:
|
[173] Wu, T.T., Yang, C.N.: Concept of nonintegrable phase factors and global formulation of Gauge fields.Phys. Rev. D 12 (1975), 3845–3857. MR 0426712, 10.1103/PhysRevD.12.3845 |
| Reference:
|
[174] Yang, C.N.: Monopoles and fiber bundles.Subnuclear Series 14 (1978), 53–84. |
| Reference:
|
[175] Yang, C.N., Mills, R.L.: Conservation of isotopic spin and isotopic gauge invariance.Phys. Rev. 96 (1954), 191–195. MR 0065437, 10.1103/PhysRev.96.191 |
| Reference:
|
[176] Yates, R.G.: Fibre bundles and supersymmetries.Comm. Math. Phys. 76 (1980), 255–268. MR 0588049, 10.1007/BF02193556 |
| Reference:
|
[177] Zardecki, A.: A formulation of supergravity based on Cartan’s connection.J. Math. Phys. 34 (1993), 1487–1496. MR 1210229, 10.1063/1.530168 |
| Reference:
|
[178] Zardecki, A.: Gravity as a Gauge theory with Cartan connection.J. Math. Phys. 29 (1998), 1661–1666. MR 0946342, 10.1063/1.528197 |
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