We will now consider the case of so-called beta-deformation.

Linearized beta-deformations transform in the following representation:


where the subindex means zero internal commutator; has .

The nonlinear solutions were studied in:

O.Aharony, B.Kol, and S.Yankielowicz, On exactly marginal deformations of SYM and type IIB supergravity on

It was shown that the renormalization of beta-deformation is again a beta-deformation, and the anomalous dimension is an expression cubic in the deformation parameter.

In our language this corresponds to the obstacle starting at . One would thing that the relevant cohomology group is:

But this cohomology group is zero, because is finite-dimensional.

What actually happens is:


where is some infinite-dimensional extension of .

We will now explain this extension.