Conformal (p, q) supergeometries in two dimensions

Abstract

We propose a superspace formulation for conformal (p, q) supergravity in two dimensions as a gauge theory of the superconformal group OSp0(p|2; ℝ) × OSp0(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 $$ \mathcal{N} $$ N -extended AdS superspace as a maximally symmetric supergeometry in the p = q ≡ $$ \mathcal{N} $$ N 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 = 2n, a possible choice of the superconformal group is SU(1, 1|n) × OSp0(q|2; ℝ), for n ≠ 2, and PSU(1, 1|2) × OSp0(q|2; ℝ), when n = 2. In general, a conformal superspace formulation is associated with a supergroup G = GL× GR, where the simple supergroups GL and GR 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 HL× HR superspace, where HL (HR) is the R-symmetry subgroup of GL (GR). 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.

Embedding formalism for (p, q) AdS superspaces in three dimensions

“…The structure of conformal supercurrents implies that conformal (p, q) supergravity [22] is characterised by two unconstrained prepotentials, Ĥ+(4−q) and Ȟ−(4−p) , which couple to the supercurrents as follows…”

“…superconformal[22], and the dimensions of Ĵ+(4−p) and J−(4−q) are − q), respectively. The functional structure of the conformal supercurrents is dictated by their top componentsĴ+(4−p) (x = , θ + ) ∝ • • • + i p(p−1) p!…”

Generalised superconformal higher-spin multiplets

“…One may argue that the problem of deriving the complete actions for N = 4 conformal supergravity in four dimensions (involving a holomorphic function of the complex scalar that parametrises an SU(1, 1)/U(1) coset space), which were constructed only a few years ago [21,22], was solved using the N = 4 conformal superspace sketched in the appendices of [22]. The N = 3 conformal superspace in four dimensions has recently been developed [23]. 3 It follows from the analysis in [24] that these definitions are equivalent to those given [11,12].…”