nLab super anti de Sitter spacetime



Riemannian geometry





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):

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)×SO(7)\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times SO(7)}\;\;\;\;
A\phantom{A}5A\phantom{A}SU(2,2|5)Spin(4,1)×SO(5)\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;
A\phantom{A}7A\phantom{A}OSp(6,2|4)Spin(6,1)×SO(4)\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;

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

In particular for d=4d=4 (see e.g. Section 3 of Kuzenko & Tartaglino-Mazzucchelli 2022):

AdS 4|4𝒩=OSp(𝒩|4;)SO(3,1)×SO(𝒩). AdS^{4\vert 4\mathcal{N}} \;=\; \frac{ OSp( \mathcal{N} \vert 4; \mathbb{R} ) } { SO(3,1) \times SO(\mathcal{N}) } \,.

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


General discussion:

Specifically on super-AdS 4AdS_4 (super-near horizon geometry of black M2-branes as in AdS4/CFT3-duality):

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 in

(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

Last revised on April 23, 2024 at 15:54:37. See the history of this page for a list of all contributions to it.