super anti de Sitter spacetime in nLab

Contents

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):

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;ℝ) bos≃Sp(4;ℝ)×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;ℝ)≃Spin(2,3)Sp(4; \mathbb{R}) \simeq Spin(2,3) [e.g. Garret 2013, §2.8].

Properties

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).

• 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×SadS \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×S 7adS_4 \times S^7 and adS 7×S 4adS_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×S 7AdS_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=11D=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 | 4 ) 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×S 7AdS_4 \times S^7 and AdS 7×S 4AdS_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)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×ℂP 3AdS_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=11D=11 supergravity and M2-brane, MSc thesis (2023) [pdf, pdf]

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

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

Specifically for AdS 4|4𝒩AdS^{4\vert 4 \mathcal{N}}:

• Nowar E. Koning, Sergei M. Kuzenko, Emmanouil S. N. Raptakis: The anti-de Sitter supergeometry revisited [arXiv:2412.03172]

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

• Ruslan Metsaev, Arkady Tseytlin, Type IIB superstring action in AdS 5×S 5AdS_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×S 5AdS_5 \times S^5 supertwistor space, JHEP 0011:024 (2000) [arXiv:hep-th/0010104, doi:10.1088/1126-6708/2000/11/024]