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  Pre-gauge-fixed superspace action of the RNS superstring

+ 1 like - 0 dislike
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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:=iT4Wd2σd2θˉDYDY,

where W is the superworldsheet, Y is a superfield on W: Y:=X+ˉθψ+12ˉθθB,
D is the 'supercovariant derivaitve': DA:=ˉθA+(ραθ)Aα,
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:=T2Wd2σhαXαX,

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
asked Nov 16, 2013 in Theoretical Physics by Jonathan Gleason (265 points) [ no revision ]

1 Answer

+ 1 like - 0 dislike

It is given in an old review by D'Hoker and Phong, with this INSPIRE entry.

A summary is given in Sec. 2.1 and Appendix B of (again) D'Hoker and Phong, hep-th/0110238.

This post imported from StackExchange Physics at 2014-08-23 05:00 (UCT), posted by SE-user Yuji
answered Nov 16, 2013 by Yuji (1,395 points) [ no revision ]

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