Conformal (p, q) supergeometries in two dimensions

Abstract

A bstract We propose a superspace formulation for conformal ( p , q ) supergravity in two dimensions as a gauge theory of the superconformal group OSp 0 ( p |2; ℝ ) × OSp 0 ( q |2; ℝ ) with a flat connection. Upon degauging of certain local symmetries, this conformal superspace is shown to reduce to a conformally flat SO( p ) × SO( q ) superspace with the following properties: (i) its structure group is a direct product of the Lorentz group and SO( p ) × SO( q ); and (ii) the residual local scale symmetry is realised by super-Weyl transformations with an unconstrained real parameter. As an application of the formalism, we describe $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -extended AdS superspace as a maximally symmetric supergeometry in the p = q ≡ $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> case. If at least one of the parameters p or q is even, alternative superconformal groups and, thus, conformal superspaces exist. In particular, if p = 2 n , a possible choice of the superconformal group is SU(1, 1| n ) × OSp 0 ( q |2; ℝ ), for n ≠ 2, and PSU(1, 1|2) × OSp 0 ( q |2; ℝ ), when n = 2. In general, a conformal superspace formulation is associated with a supergroup G = G L × G R , where the simple supergroups G L and G R can be any of the extended superconformal groups, which were classified by Günaydin, Sierra and Townsend. Degauging the corresponding conformal superspace leads to a conformally flat H L × H R superspace, where H L ( H R ) is the R -symmetry subgroup of G L ( G R ). Additionally, for the p , q ≤ 2 cases we propose composite primary multiplets which generate the Gauss-Bonnet invariant and supersymmetric extensions of the Fradkin-Tseytlin term.

Topics & Concepts