Four twisted differential operators for the N=4N=4 superconformal algebra

As it was shown by Vafa, Witten and Marcus, the N=4N=4, d=4d=4 Yang–Mills super-Poincaré Lagrangian can be twisted and its quantum field theory can be interpreted as a topological theory. It was shown afterward that, using twisted representations for the scalar and spinor fields of this theory, one can find so-called twisted representations of the N=4N=4, d=4d=4 Yang–Mills super-Poincaré supersymmetry, which are small enough to close off-shell and big enough to fully determine the theory. This allows for instance proofs of various finiteness properties of the N=4N=4 theory that only rely on locality properties and a standard use of Ward identities. Moreover, the geometrical aspects of these transformations are quite appealing, in a way that generalizes non-trivially the case of the N=2N=2, d=4d=4 supersymmetry. Here, we show that, surprisingly, the N=4N=4, d=4d=4 Yang–Mills conformal supersymmetry also exhibits a very simple sub-sector described by four differential operators. The invariance under this subalgebra is big enough to determine the N=4N=4 theory. There is in fact a very closed link between the twisted vector supersymmetry of the super-Poincaré algebra and both twisted scalar twisted supersymmetries of the superconformal algebra, which explains the existence of supersymmetric observables. Some attempts are done to interpret these differential operators.