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
SRNS:=iT4∫Wd2σd2θˉDY⋅DY,
where
W is the superworldsheet,
Y is a superfield on
W:
Y:=X+ˉθψ+12ˉθθB,
D is the 'supercovariant derivaitve':
DA:=∂∂ˉθA+(ραθ)A∂α,
ρα are generators of the
(−,+) Clifford algebra:
{ρα,ρβ}=2ηαβ,
and the bar denotes the Dirac conjugate.
On the other hand, for the bosonic string, we have the Polyakov action:
SP:=−T2∫Wd2σ√−h∇αX⋅∇αX,
where
hαβ is the metric on the worldsheet
W and
∇α the corresponding Levi-Civita covariant derivative (which for scalar fields happens to agree with just the usual partial derivative). If we take
hαβ=ηαβ (conformal gauge), then this reduces to the Bosonic part (ignoring the auxilary field
B) of
SRNS.
I was wondering: what is the appropriate generalization of SRNS 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 supercovariant1 derivative? This generalization should be analogous to the pre-gauge-fixed form of SP 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.
This post imported from StackExchange Physics at 2014-08-23 05:00 (UCT), posted by SE-user Jonathan Gleason