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 ${\mathsf{OSp}}_0(p|2;\mathbb{R})\times {\mathsf{OSp}}_0(q|2;\mathbb{R})$ with a flat connection. Upon degauging of certain local symmetries, this conformal superspace is shown to reduce to a conformally flat $\mathsf{SO}(p)\times \mathsf{SO}(q)$ superspace with the following properties: (i) its structure group is a direct product of the Lorentz group and $\mathsf{SO}(p)\times \mathsf{SO}(q)$; 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 $\mathcal{N}$-extended AdS superspace as a maximally symmetric supergeometry in the $p=q\equiv \mathcal{N}$ 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 $\mathsf{SU}(1,1|n)\times {\mathsf{OSp}}_0(q|2;\mathbb{R})$, for $n\ne 2$, and $\mathsf{PSU}(1,1|2)\times {\mathsf{OSp}}_0(q|2;\mathbb{R})$, when $n=2$. In general, a conformal superspace formulation is associated with a supergroup $G=G_L\times G_R$, 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 $H_L\times H_R$ superspace, where $H_L$ ($H_R$) is the $R$-symmetry subgroup of $G_L$ ($G_R$). Additionally, for the $p,q\le 2$ cases we propose composite primary multiplets which generate the Gauss-Bonnet invariant and supersymmetric extensions of the Fradkin-Tseytlin term.

arXiv comment42 pages; V2: 49 pages, references, comments and a new appendix added

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Cited by in corpus (4)

• On curvature-squared invariants of minimal five-dimensional supergravity from superspace
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• Embedding formalism for $(p,q)$ AdS superspaces in three dimensions
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