Deformations of vs deformations of super-Yang-Mills
Consider single trace deformations of supersymmetric Yang-Mills theory (SYM):
where is a single-trace operator.
According to AdS/CFT Aharony:1999ti, they correspond to some deformations of the Type IIB superstring theory
on . Let us restrict ourselves to infinitesimal deformations, i.e. compute
only to the first order in the deformation parameter . Besides , there are two
other parameters:
the number of colors
and the Yang-Mills coupling constant
Consider .
Then, there are two opposite limits: the limit of free super Yang-Mills theory (SYM) and the strong coupling limit . In the weak coupling limit we can do perturbative calculations in SYM, and in the strong coupling limit we can use the superstring theory on the classical supergravity (SUGRA) background .
Both SYM theory and superstring on are invariant under the superconformal group
, and it is natural to ask how the deformations of the form (1) transform
under this group. In particular, some deformations of the form (1)
transform in finite-dimensional representations Mikhailov:2011af.
As an example of a finite-dimensional representation, consider of Eq. (1) of the form:
where are the scalar fields of the -super-Yang-Mills theory. Consider
the linear space of all deformations obtained from Eq. (2) by acting with all possible
polynomials of generators of . In the free theory, this gives a finite-dimensional representation.
Let us call it .
It was argued in Mikhailov:2017uoh that this representation is not deformed when turning on , i.e. is the same in free and interacting theory.