Conformal $(p,q)$ supergeometries in two dimensions

TL;DR

This paper develops a superspace formulation for conformal (p,q) supergravity in two dimensions, describing its structure, reductions, and applications to AdS superspaces, with new superconformal groups and primary multiplets.

Contribution

It introduces a novel superspace approach for 2D conformal (p,q) supergravity, including new superconformal groups, reductions to flat superspaces, and primary multiplets for invariants.

Findings

Superspace formulation for conformal (p,q) supergravity in 2D.

Reduction to conformally flat superspaces with super-Weyl transformations.

Construction of primary multiplets generating invariants like Gauss-Bonnet.

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 ${\cal N}$-extended AdS superspace as a maximally symmetric supergeometry in the $p=q \equiv \cal N$ 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 $n \neq 2$, 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 G\"unaydin, 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 $p,q \leq 2$ cases we propose composite primary multiplets which generate the Gauss-Bonnet invariant and supersymmetric extensions of the Fradkin-Tseytlin term.