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The orthosymplectic supergroup is the sub-supergroup of the general linear supergroup on those elements which preserve the canonical graded-symmetric bilinear form on the super vector space , i.e. the form which is the canonical symmetric bilinear form on the even elements , is the canonical symplectic form on the odd elements in and is zero on mixed pairs of elements.
The corresponding super Lie algebras are called the orthosymplectic Lie algebras . Over a field of characteristic zero these constitute the infinite - and -series in the classification of simple super Lie algebras. They are closely related to superconformal symmetry (e.g. D’Auria, Ferrara & Lledo-Varadarajan 2p00), see at supersymmetry – Classification – Superconformal symmetry.
is a limiting case of the M-theory super Lie algebra (Fernandez-Izquierdo-Olmo 15).
Further relation also to the type II supersymmetry algebra had been discussed in (Bergshoess-Proeyen 00).
The M-theory super Lie algebra is actually a sub-super Lie algebra of (vanHolten-VanProeyen 82, Bars-Deliduma-nMinic 99, (11)-(15), West 00, section 7)
In fact is the smallest simple supergroup that contains the M-theory super Lie algebra. Moreover, it is generated from its M-theory sub-algebra and the special conformal transformations (Bars-Deliduman-Minic 99, p. 5)
The super anti de Sitter spacetimes are quotients of orthosymplectic super Lie groups:
super anti de Sitter spacetime | ||
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4 | D’Auria & Fré 1983 | |
5 | ||
7 |
Victor Kac, pages 39-40 of: A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 1 (1977) 31-64 [euclid:1103900590, doi:10.1007/BF01609166]
Vladimir Rittenberg, (21)-(22) in: A guide to Lie superalgebras, in P. Kramers, A. Rieckers (eds.): Group Theoretical Methods in Physics, Lecture Notes in Physics 79 Springer (1978) 3-21 [doi:10.1007/3-540-08848-2_1, pdf]
Manfred Scheunert, chapter II, 4.3.A of: The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716, Springer (1979) [doi:10.1007/BFb0070929]
Richard Joseph Farmer, Orthosymplectic superalgebras in mathematics and science, PhD Thesis (1984) [pdf]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, sections II.2.6, II.3.2-3, II.5, and V.4.4 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [ch II.2: pdf, ch II.3: pdf, ch II.5: pdf, ch V.4: pdf]
Riccardo D'Auria, S. Ferrara, M. A. Lledó, Veeravalli Varadarajan, Spinor Algebras, J.Geom.Phys. 40 (2001) 101-128 (arXiv:hep-th/0010124)
N. A. Gromov, I. V. Kostyakov, V. V. Kuratov, Cayley-Klein contractions of orthosymplectic superalgebras (arXiv:hep-th/0110257)
Steven V Sam, Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon-Northcott complex (arXiv:1312.2255)
G.I. Lehrer, R.B. Zhang, The second fundamental theorem of invariant theory for the orthosymplectic supergroup (arXiv:1407.1058)
Kevin Coulembier, The orthosymplectic supergroup in harmonic analysis, J. Lie Theory 23 (2013) 55-83 [arXiv:1202.0668]
The identification of super with a coset space of is due to:
Further discussion:
Igor Bandos, Jerzy Lukierski, Christian Preitschopf, Dmitri Sorokin: supergroup manifolds, superparticles and supertwistors, Phys. Rev. D 61 (2000) 065009 [arXiv:hep-th/9907113]
Eric Bergshoeff, Antoine Van Proeyen: The many faces of , Class. Quantum Grav. 17 (2000) 3277-3303 [arXiv:hep-th/0003261, doi:10.1088/0264-9381/17/16/312]
Horatiu Nastase, Towards a Chern-Simons M theory of (arXiv:hep-th/0306269)
Luca Carlevaro: Three approaches to M-theory, PhD thesis (2006) [hdl:123456789/16186, pdf, spire:1253257]
Itzhak Bars, Dmitry Rychkov. Background Independent String Field Theory [arXiv:1407.4699]
(on string field theory with -symmetry)
J. J. Fernandez, José M. Izquierdo, M. A. del Olmo, Contractions from to the M-theory superalgebra extended by additional fermionic generators, Nuclear Physics B 897 (2015) 87-97 [arXiv:1504.05946]
(relating to the M-theory super Lie algebra)
Jan-Willem van Holten, Antoine Van Proeyen, Supersymmetry Algebras in , , , J.Phys. A15 (1982) 3763 (spire:177060)
Itzhak Bars, C. Deliduman, D. Minic, Lifting M-theory to Two-Time Physics, Phys.Lett. B457 (1999) 275-284 (arXiv:hep-th/9904063)
Itzhak Bars, 2T Physics Formulation of Superconformal Dynamics Relating to Twistors and Supertwistors, Phys.Lett. B483 (2000) 248-256 (arXiv:hep-th/0004090)
Peter West, Hidden Superconformal Symmetry in M Theory, JHEP 0008:007, 2000 (arXiv:hep-th/0005270)
Appearance of sl(2)-modular functor is found in the seemingly plain sl(2)-WZW model for fractional level (see there) in:
On the representation theory (supermultiplets, singleton representations) of orthosymplectic groups
Hermann Nicolai, Ergin Sezgin, Singleton representations of , Physics Letters B 143 4-6 16 (1984) 389-395
Murat Günaydin: Unitary supermultiplets of and M theory, Nucl. Phys. B 528 (1998) 432-450 [doi:10.1016/S0550-3213(98)00393-9, arXiv:hep-th/9803138]
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