Beta-deformation
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 
.
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 
https://arxiv.org/abs/hep-th/0205090
SYM and type IIB supergravity
on
https://arxiv.org/abs/hep-th/0205090It 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.
is finite-dimensional.What actually happens is:
where 
is some infinite-dimensional extension of 
.
is some infinite-dimensional extension of .We will now explain this extension.