Contents

Idea

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):

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

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

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

Properties

Explicit formulas for super Cartan connections for $AdS_4 \times S^7$ and $AdS_7 \times S^4$ 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).

Related concepts

• super anti de Sitter group

• super Minkowski spacetime

• AdS/CFT correspondence, AdS/QCD correspondence

References

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

General discussion:

• 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]

• Renata Kallosh, J. Rahmfeld, Arvind Rajaraman, Near Horizon Superspace, JHEP 9809:002 (1998) [arXiv:hep-th/9805217, doi:10.1088/1126-6708/1998/09/002]

• Piet Claus, Renata Kallosh, Superisometries of the $adS \times S$ Superspace, JHEP 9903:014 (1999) [arXiv:hep-th/9812087, doi:10.1088/1126-6708/1999/03/014]

• Piet Claus, Super M-brane actions in $adS_4 \times S^7$ and $adS_7 \times S^4$, Phys. Rev. D 59 (1999) 066003 [arXiv:hep-th/9809045, doi:10.1103/PhysRevD.59.066003]

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

• Sergei Kuzenko, Gabriele Tartaglino-Mazzucchelli, Supertwistor realisations of AdS superspaces, The European Physical Journal C 82 2 (2022) 146 [doi:10.1140/epjc/s10052-022-10072-y, arXiv:2108.03907]

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

• 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]

• Mike Duff, Bengt Nilsson, Christopher Pope, pp. 33 in: Kaluza-Klein supergravity, Physics Reports 130 1–2 (1986) 1-142 [spire:229417, doi:10.1016/0370-1573(86)90163-8]

  (no discussion of superspace, but of the $OSp(8 \vert 4)$-supersymmetry)

• Castellani, D’Auria & Fré 1991, §3.1

• Bernard de Wit, Kasper Peeters, Jan Plefka, Alexander Sevrin, The M-theory two-brane in $AdS_4 \times S^7$ and $AdS_7 \times S^4$, Physics Letters B 443 1-4 (1998) 153-158 [doi:10.1016/S0370-2693(98)01340-9, inspire:474621, arXiv:hep-th/9808052]

• Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, §4.1 in: The $Osp(8|4)$ singleton action from the supermembrane, Nucl. Phys. B 542 (1999) 157-194 [arXiv:hep-th/9807115, doi:10.1016/S0550-3213(98)00765-2]

• Jaume Gomis, Dmitri Sorokin, Linus Wulff, The complete $AdS_4 \times \mathbb{C}P^3$ superspace for the type IIA superstring and D-branes, JHEP 0903:015 (2009) [arXiv:0811.1566]

Review:

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

Specifically for the superstring on super-$AdS_5 \times S^5$:

• Igor A. Bandos: Superembedding approach to superstring in $AdS_5 \times S^5$ superspace, in: Fundamental Interactions (2009) 303-334 [arXiv:0812.0257, doi:10.1142/9789814277839_0018]

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.B 533 (1998) 109-126 [arXiv:hep-th/9805028]

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:

• Radu Roiban, Warren Siegel, Superstrings on $AdS_5 \times S^5$ supertwistor space, JHEP 0011:024 (2000) [arXiv:hep-th/0010104, doi:10.1088/1126-6708/2000/11/024]