Since this question is still open and therefore not definitely answerable at present, I save the valuable discussion of the topic in the comments as an answer such that it does not get lost:
This is just an accident of 10 dimensions--- there is too much supersymmetry to have a full SUSY superspace. It's a very good question, but research level, if you answer it fully, everyone will breathe easier. – Ron Maimon Apr 12 at 2:11 3
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There are superspace formulations for 4D N=4 theories, the problem is that they only are on-shell formulations. The N=4 multiplet contains the N=2 hypermultiplet and there is a no-go theorem saying that there is no off-shell formulation with a finite number of auxiliary fields. Thus you get projective and harmonic superspaces with infinite numbers of auxiliary fields. This works for N=2, but in N=4 all constraints to reduce the unconstrained superfields down to the physical multiplets force the fields on-shell. – Simon Apr 12 at 3:47 3
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And as @Ron says, the reason why it is so difficult to construct such formulations is an open research-level question. If the reason was known, then we'd either have a workable N=4 superspace formulation or a no-go theorem by now... – Simon Apr 12 at 3:48 3
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Dear Dilaton, good question. Ron and Simon have already answered to some extent and I will only offer a different extent, extending Ron's comment in particular. If you want to make N=1 SUSY in 4D i.e. 4 real supercharges manifest, you need 4 superspace fermionic coordinates. With 16 supercharges, you would probably need at least 16 fermionic coordinates in the superspace but then the fields would have 216=256 components which is pretty high give that you only need 16 on-shell components only. Most of the component fields would have to be auxiliary, linked to deritives of others, etc. Hard – Luboš Motl Apr 12 at 5:37 4
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There is also an interesting twistor-like transform for the 10D super Yang-Mills by a guy called Witten – Luboš Motl Apr 12 at 5:39
Thanks guys for these valuable comments and the cool links therein. I would "like" and appreciate them as "partial" answers (since as you say there is no full answer on this yet) too ... :-). – Dilaton Apr 12 at 8:45
@LubošMot l must admit that twistors are one of my black holes of ignorance (I did not get it from Roger Penrose's "Road to Reality") :-/ ... So I'm checking from time to time if I can find a nice "pedagogical" introduction to this on TRF ;-) – Dilaton Apr 12 at 8:50 1
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@Dilaton: This is the problem with research level stuff, I can't answer because I don't feel confident enough in my biases about what the answer could or should be to put them in writing, and I would mention a bunch of things that I tried and didn't work to answer this, and are not interesting, and I think everyone else is hesitant to answer too for similar reasons. Maybe you could copy the comment thread into an answer box, and accept your own answer. – Ron Maimon 2 hours ago 1
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I mean, if you want a little more on this--- there is the question of whether superspace is fundamental in the first place--- it's just a trick for writing multiplets in a way that takes the SUSY off shell naturally, but the physical reasoning has always eluded me. I tried Nicolai maps as an alternative, but it never worked, and it always is tantalizingly close to working, and I tried learning harmonic superspace for on-shell N=4, but although it is correct, its so annoyingly complicated to work with! And the S-matrix is simple, so there's a better language out there, I don't know what. – Ron Maimon 1 hour ago
Thanks @Ron Maimon, that is a good idea to save the discussion into an answer. I`m somehow intrigued too by the question if superspace itself could have some physical meaning ... – Dilaton 2 mins ago
Even though the question is still open, it could probably nevertheless be worthwhile to know what you tried and why it did not work. I mean similar to some kind of "null results" who can be interesting too ...? – Dilaton
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