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

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.