nLab super anti de Sitter spacetime

Idea

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

$\phantom{A}$$d$$\phantom{A}$	$\phantom{A}$super anti de Sitter spacetime$\phantom{A}$
$\phantom{A}$4$\phantom{A}$	$\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times SO(7)}\;\;\;\;$
$\phantom{A}$5$\phantom{A}$	$\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;$
$\phantom{A}$7$\phantom{A}$	$\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;$

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

Notice that the de Sitter spacetime (not anti-de Sitter) does not have a standard extension to supergeometry, but see arXiv:1610.01566.

Related concepts

• super anti de Sitter group

• super Minkowski spacetime

• AdS/CFT correspondence, AdS/QCD correspondence

References

General discussion includes

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, sections II.2.6, II.3.2 and II.3.3 in Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Antoine Van Proeyen, sections 4.5, 4.7 of Tools for supersymmetry (arXiv:hep-th/9910030)

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

• Ruslan Metsaev, Arkady Tseytlin, Type IIB superstring action in $AdS_5 \times S^5$ background, Nucl. Phys.B533:109-126, 1998 (arXiv:hep-th/9805028)

However, a supersymmetric trivialization of this cocycle seems to have been obtained in

• Radu Roiban, Warren Siegel, Superstrings on $AdS_5 \times S^5$ supertwistor space, JHEP 0011:024, 2000 (arXiv:hep-th/0010104)

(according to arxiv:1808.04470, p. 5 and equation (5.5), but check).

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