For 4 real SUSY generators, the superspace formalism has been worked out a long time ago. For 8 real SUSY generators, some brilliant theoreticians have worked out the details of harmonic superspace. The major surprise is we have to enlarge superspace by taking the product with $S^2$. When we go down to the original superspace without the $S^2$ factor, we end up with infinitely many harmonics, almost all of which are auxiliary.

Is it worthwhile to construct a manifestly supersymmetric superspace formalism for 16 or 32 real SUSY generators? The number of auxiliary fields will be enormous compared to the number of dynamical fields, but will this effort pay off?

Hybrid formalisms like 4D superspace formalism in higher dimensions are unsatisfactory as they don't treat all bosonic dimensions on an equal footing.

  • $\begingroup$ What is the definition of superspace? Is it the same as that of supermanifold? $\endgroup$
    – Marek
    Feb 19 '11 at 18:04

Win, it's been tried. For ordinary supersymmetry in flat space with 16 or even 32 supercharges, the superspace description, even if it could work, would surely not be worth the effort. Note that for 32 supercharges, the full multiplet has 4 billion components (4,294,967,296 if you want me to be accurate), much like the number of IPv4 addresses. They managed to run out after 40 years but no one wants to do calculations for 40 years.

The reason why the super-extended superspaces are not useful is that the superspace formulation of theories with 4 or 8 supercharges is helpful because it

  • makes supersymmetry manifest but also
  • allows us to formulate a large number of physically distinct theories.

If you managed to define a superspace for 16 or 32 supercharges, the first condition would still be satisfied. However, the second one would not be. The massless theories with 16 or 32 supercharges are the maximally supersymmetric Yang-Mills theory and maximally supersymmetric supergravity, respectively.

Supersymmetry in those two cases is so constraining that only the gauge group may be adjusted in the 16-supercharge Yang-Mills case; and nothing at all can be adjusted in the 32-supercharge case of supergravity.

That's very different from the case of four supercharges, e.g. in $N=1$, where superspace allows you to define a general superpotential and Kähler potential (functions of an arbitrary number of fields - chiral and vector multiplets - you may start with), or $N=2$ where you have a rather adjustable prepotential etc. There's no counterpart of these choices if you have 16 or more supercharges.

So it's not useful although it could be a mathematical curiosity. One would have to decompose the theory in the components, anyway. And it would always be the same theory.

Counterexamples could involve theories outside flat space - where a high degree of supersymmetry could lead to some surprises. Harmonic superspaces involve bosonic coordinates as well and may produce nontrivial theories, e.g. gauged supergravity. And exceptions could exist for massive multiplets but in those highly supersymmetric theories, massive states are constrained, too. In particular, $N=4$, $d=4$ gauge theory is conformal so it has no preferred mass scale. And massive states in maximal supergravity are inevitably black hole (or black brane) microstates of a sort, so field theory that treats them as excitations is not a good description, anyway.


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