# Pre-gauge-fixed superspace action of the RNS superstring

When writing down the the action of the RNS superstring in superspace, all of the sources I have checked (BBS, GSW, Polchinski) seem to just write down the action in conformal gauge, that is $$S_{\text{RNS}}:=\mathrm{i}\, \frac{T}{4}\int _W\mathrm{d}^2\sigma \mathrm{d}^2\theta \, \bar{D}Y\cdot DY,$$ where $W$ is the superworldsheet, $Y$ is a superfield on $W$: $$Y:=X+\bar{\theta}\psi +\frac{1}{2}\bar{\theta}\theta B,$$ $D$ is the 'supercovariant derivaitve': $$D_A:=\frac{\partial}{\partial \bar{\theta}^A}+(\rho ^\alpha \theta )_A\partial _\alpha,$$ $\rho ^\alpha$ are generators of the $(-,+)$ Clifford algebra: $$\{ \rho ^\alpha ,\rho ^\beta \}=2\eta ^{\alpha \beta},$$ and the bar denotes the Dirac conjugate.

On the other hand, for the bosonic string, we have the Polyakov action: $$S_{\text{P}}:=-\frac{T}{2}\int _W\mathrm{d}^2\sigma \, \sqrt{-h}\nabla _\alpha X\cdot \nabla ^\alpha X,$$ where $h_{\alpha \beta}$ is the metric on the worldsheet $W$ and $\nabla _\alpha$ the corresponding Levi-Civita covariant derivative (which for scalar fields happens to agree with just the usual partial derivative). If we take $h_{\alpha \beta}=\eta _{\alpha \beta}$ (conformal gauge), then this reduces to the Bosonic part (ignoring the auxilary field $B$) of $S_{\text{RNS}}$.

I was wondering: what is the appropriate generalization of $S_{\text{RNS}}$ to a theory defined on a supermanifold with a 'supermetric'? For that matter, what is the right notion of a supermetric on a supermanifold and do we have an analogous Fundamental Theorem of Super-Riemannian Geometry that, given a supemetric, gives us a canonical supercovariant$^1$ derivative? This generalization should be analogous to the pre-gauge-fixed form of $S_{\text{P}}$ given above in which the metric and all covariant derivatives appear explicitly.

$^1$ While $D$ is called the "supercovariant derivative", it clearly cannot be the right notion, at least not in general, because it makes no reference to a supermetric.