Explaining Pure Spinor Superspace

In the pure spinor formalism for the superstring and supermembrane, supersymmet-ric invariants are constructed by integrating over five θ's in d=10 and over nine θ's in d=11. This pure spinor superspace is easily explained using the superform (or " ecto-plasm ") method developed by Gates and collaborators, and generalizes the standard chiral superspace in d=4. The ectoplasm method is also useful for constructing d=10 and d=11 supersymmetric invariants in curved supergravity backgrounds. The conventional method for constructing supersymmetric invariants is to integrate superfields over a superspace which contains both x m and θ α variables. The number of θ's which must be integrated depends both on the spacetime dimension and on the constraints satisfied by the superfields. For example, d=4 supersymmetric invariants can be constructed either by using real superfields and integrating over four θ's, or by using chiral superfields and integrating over two θ's. Although it is somewhat non-trivial to generalize these d=4 supersymmetric invariants in a curved supergravity background, this can be done by inserting the appropriate constrained supervielbeins into the superspace integral. This conventional method for constructing supersymmetric invariants is less useful in higher spacetime dimensions which involve more θ's. For example, the construction of d=10 super-Poincaré invariant expressions using unconstrained superfields would require integration over 16 θ's, which means that the supersymmetric invariants typically involve terms with eight spacetime derivatives. Although one can try to define constrained d=10 superfields which allow integration over fewer than 16 θ's, finding an appropriate set of constraints is not easy. Furthermore, if one finds a suitable set of constraints, it is not obvious how to generalize them in a curved supergravity background. Over the last six years, an appropriate set of constraints for d=10 and d=11 superfields has been discovered using the pure spinor formalism for the superstring and supermem-brane [1][2]. Using the constraints coming from these pure spinor formalisms, d=10 and d=11 supersymmetric invariants in a flat background have been constructed involving as few as two spacetime derivatives. These supersymmetric invariants naturally arise as on-shell scattering amplitudes in the pure spinor approach.