nLab orthosymplectic super Lie algebra




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

The corresponding super Lie algebras are called the orthosymplectic Lie algebras 𝔬𝔰𝔭(N|2p)\mathfrak{osp}(N|2p). Over a field of characteristic zero these constitute the infinite BB- and DD-series in the classification of simple super Lie algebras. They are closely related to superconformal symmetry (e.g. D’Auria-Ferrara-Lledo-Varadarajan 00), see at supersymmetry – Classification – Superconformal symmetry.


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



In superstring theory

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

Further discussion:

Appearance of 𝔬𝔰𝔭(1|2)×\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 𝔰𝔩 2^\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

The representation theory (singleton representations) of the orthosymplectic group is discussed in

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

Last revised on April 26, 2024 at 10:27:36. See the history of this page for a list of all contributions to it.