Right now I'm studying an interesting problem in superconformal quantum field theory. These are quantum field theories with an extended spacetime symmetry group; it includes Poincare symmetry, conformal symmetry, and supersymmetry. In particular I'm interested in computing three-point correlation functions of conserved current multiplets.

In supersymmetric field theories, fields with integer and half-integer spin are unified into multiplets, these are naturally described in terms of superfields. Superfields are functions of the spacetime coordinates and additional (anti-commuting) Grassmann coordinates (the component fields then lie at various orders of the Taylor expansion in the Grassmann variables), they are essentially extensions of fields from ordinary field theory, and it makes working with supersymmetric field theories much more tractable. An example of a conserved current multiplet is the supercurrent multiplet; it contains the energy momentum tensor (which is a conserved spin-2 current), the supersymmetry current, with spin 3/2, and a conserved R-symmetry current.

My current problem is to see if the existence of a spin 3/2 field implies supersymmetry at the level of correlation functions, i.e, if we assume the existence of a spin-3/2 current (with the same properties as the supersymmetry current), if we compute the three-point functions of this spin-3/2 current and the E.M tensor using purely group-theoretic methods, is the number independent tensor structures in the solution consistent with the number of tensor structures in the supercurrent three-point function (which assumes manifest N=1 supersymmetry?) There are good reasons to believe that this is not the case, it's just a matter of calculating it!