nLab super anti de Sitter spacetime



Riemannian geometry





A supergeometric analog of anti-de Sitter spacetime (times an internal space). By the discussion at supersymmetry – Classification – Superconformal and super anti de Sitter symmetry this includes the following coset superspacetimes (super Klein geometries):

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\vert 4)}{Spin(3,1) \times \mathrm{O}(7)}\;\;\;\;
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)}\;\;\;\;

(Here Osp(|)Osp(-\vert-) denotes orthosymplectic super Lie groups.)

E.g. for d=4d=4 this identification [D’Auria & Fré 1983] relies on the fact that OSp(8|4;) bosSp(4;)×O(8)OSp(8 \vert 4; \mathbb{R})_{bos} \simeq Sp(4;\mathbb{R}) \times O(8) and then on the exceptional isomorphism Sp(4;)Spin(2,3)Sp(4; \mathbb{R}) \simeq Spin(2,3) [e.g. Garret 2013, §2.8].


On the other hand, de Sitter spacetimes (instead of anti-de Sitter) do not have a standard extensions to supergeometry; but see arXiv:1610.01566.


Explicit formulas for super Cartan connections for AdS 4×S 7AdS_4 \times S^7 and AdS 7×S 4AdS_7 \times S^4 are given in dWPPS 98, p. 156, following Kallosh, Rahmfeld & Rajaraman 1998 and Claus & Kallosh 1999, see also Claus 1998 (reviewed in Wang 2023, §4.4).


See also the references at Green-Schwarz sigma-model – References – AdS backgrounds.

General discussion:

Specifically concerning super-AdS 4×S 7AdS_4 \times S^7 (super-near horizon geometry of black M2-branes as in AdS4/CFT3-duality):


  • Zihan Wang, A review on D=11D=11 supergravity and M2-brane, MSc thesis (2023) [pdf, pdf]

The super 3-cocycle for the Green-Schwarz superstring on SU(2,2|5)Spin(4,1)×SO(5)\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)} is originally due to

However, a supersymmetric trivialization of this cocycle seems to have been obtained (according to arxiv:1808.04470, p. 5 and equation (5.5), but check) in:

Last revised on June 25, 2024 at 15:03:09. See the history of this page for a list of all contributions to it.