nLab orthosymplectic super Lie algebra

Idea

The orthosymplectic supergroup $OSp(N|2p)$ is the sub-supergroup of the general linear supergroup $GL(N|2p)$ on those elements which preserve the canonical graded-symmetric bilinear form on the super vector space $\mathbb{R}^{N|2p}$, i.e. the form which is the canonical symmetric bilinear form on the even elements $\mathbb{R}^{N}$, is the canonical symplectic form on the odd elements in $\mathbb{R}^{0|2p}$ and is zero on mixed pairs of elements.

The corresponding super Lie algebras are called the orthosymplectic Lie algebras $\mathfrak{osp}(N|2p)$. Over a field of characteristic zero these constitute the infinite $B$- and $D$-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.

Examples

• $\mathfrak{osp}(1\vert 32)$ 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 $\mathfrak{osp}(1\vert 64)$ (vanHolten-VanProeyen 82, Bars-Deliduma-nMinic 99, (11)-(15), West 00, section 7)

  In fact $OSp(1\vert 64)$ 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:

$\phantom{A}$$d$$\phantom{A}$	$\phantom{A}$super anti de Sitter spacetime$\phantom{A}$
$\phantom{A}$4$\phantom{A}$	$\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times \mathrm{O}(7)}\;\;\;\;$	D’Auria & Fré 1983
$\phantom{A}$5$\phantom{A}$	$\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times \mathrm{O}(5)}\;\;\;\;$
$\phantom{A}$7$\phantom{A}$	$\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times \mathrm{O}(4)}\;\;\;\;$

Related concepts

• orthogonal Lie algebra

References

General

• 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]

In superstring theory

The identification of super $AdS_4\times S^7$ with a coset space of $OSp(8 \vert 4)$ is due to:

• Riccardo D'Auria, Pietro Fré: Spontaneous generation of symmetry in the spontaneous compactification of $D=11$ supergravity, Physics Letters B 121 2–3 (1983) 141-146 [doi:10.1016/0370-2693(83)90903-6]

Further discussion:

• Igor Bandos, Jerzy Lukierski, Christian Preitschopf, Dmitri Sorokin: $OSp$ supergroup manifolds, superparticles and supertwistors, Phys. Rev. D 61 (2000) 065009 [arXiv:hep-th/9907113]

• Eric Bergshoeff, Antoine Van Proeyen: The many faces of $OSp(1|32)$, 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 $OSp(1\vert 32) \times OSp(1\vert 32)$ (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 $OSp(d\vert 2)$-symmetry)

• J. J. Fernandez, José M. Izquierdo, M. A. del Olmo, Contractions from $osp(1|32) \oplus osp(1|32)$ 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, $N=1$ Supersymmetry Algebras in $D=2$, $D=3$, $D=4$ $MOD-8$, 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 $\mathfrak{osp}(1|2) \times$sl(2)-modular functor is found in the seemingly plain sl(2)-WZW model for fractional level (see there) in:

• Boris Feigin, Feodor Malikov, Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level, p. 357-405 in: Loday, Stasheff, Voronov (eds.) Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) [arXiv:q-alg/9511011, ams:conm-202]

Representation theory

On the representation theory (supermultiplets, singleton representations) of orthosymplectic groups

• Hermann Nicolai, Ergin Sezgin, Singleton representations of $Osp(N,4)$, Physics Letters B 143 4-6 16 (1984) 389-395

• Murat Günaydin: Unitary supermultiplets of $OSp(1/32,R)$ and M theory, Nucl. Phys. B 528 (1998) 432-450 [doi:10.1016/S0550-3213(98)00393-9, arXiv:hep-th/9803138]