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 .

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

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.