The anti-de Sitter supergeometry revisited

A bstract In a supergravity framework, the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -extended anti-de Sitter (AdS) superspace in four spacetime dimensions, $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>AdS</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mfenced> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msup> </mml:math> , is a maximally symmetric background that is described by a curved superspace geometry with structure group SL(2, ℂ ) × $$ \textrm{U}\left(\mathcal{N}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> <mml:mfenced> <mml:mi>N</mml:mi> </mml:mfenced> </mml:math> . On the other hand, within the group-theoretic setting, $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>AdS</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mfenced> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msup> </mml:math> is realised as the coset superspace $$ \textrm{O}\textrm{Sp}\left(\left.\mathcal{N}\right|4;\mathbb{R}\right)/\left[\textrm{SL}\left(2,\mathbb{C}\right)\times \textrm{O}\left(\mathcal{N}\right)\right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>OSp</mml:mi> <mml:mfenced> <mml:mrow> <mml:mfenced> <mml:mi>N</mml:mi> </mml:mfenced> <mml:mn>4</mml:mn> </mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mfenced> <mml:mo>/</mml:mo> <mml:mfenced> <mml:mrow> <mml:mi>SL</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℂ</mml:mi> </mml:mfenced> <mml:mo>×</mml:mo> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mi>N</mml:mi> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> , with its structure group being SL(2, ℂ ) × $$ \textrm{O}\left(\mathcal{N}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mi>N</mml:mi> </mml:mfenced> </mml:math> . Here we explain how the two frameworks are related. We give two explicit realisations of $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>AdS</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mfenced> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msup> </mml:math> as a conformally flat superspace, thus extending the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 and $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 results available in the literature. As applications, we describe: (i) a two-parameter deformation of the $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>AdS</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mfenced> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msup> </mml:math> interval and the corresponding superparticle model; (ii) some implications of conformal flatness for superconformal higher-spin multiplets and an effective action generating the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 super-Weyl anomaly; and (iii) κ -symmetry of the massless AdS superparticle.