# What are the limitations of the superspace formalism?

Just from reading this slightly technical introduction to supersymmetry and watching these Lenny Susskind lectures, I thought that the Lagrangian of any "reasonable" supersymmetric theory can always by derived from the superfield formalism; such that the F term of the superpotential contains the mass and interaction terms and the D term of $\Phi^{\dagger}\Phi$ describes the kinetic terms or free part.

But then I read in this article (with the main topic about N=4 SYM theories) as an "aside note", that there exists no (known) superspace for D=10, N=1 YM theories described by the Lagrangian

$$L = \mathrm{tr} \left[ -\frac{1}{4}F_{\mu\nu} F^{\mu\nu} + i\bar{\Psi}D^{\mu}\gamma_{\mu}\Psi \right]$$

for example.

My questions now are:

Is there an "easy" or "intuitive" (meaning such that I can get it :-P...) way to understand why in this case there is no (known) superspace for this theory (such that the Lagrangian can not be derived from the methods mentioned above?). Or more generally what are the limitations of the superspace formalism; for which kind of theories does it work and under what conditions is it not applicable?

• 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. Apr 12, 2012 at 2:11
• 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. Apr 12, 2012 at 3:47
• 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... Apr 12, 2012 at 3:48
• 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 $2^{16}=256$ 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 Apr 12, 2012 at 5:37
• There is also an interesting twistor-like transform for the 10D super Yang-Mills by a guy called Witten, sciencedirect.com/science/article/pii/0550321386900908 Apr 12, 2012 at 5:39

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
up voted

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
up voted

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
up voted

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
up voted

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
up voted

@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
up voted

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