Linear dilaton
First of all, the “most leading” deformation actually has , i.e.
.
The action is not deformed.
What does get deformed is the BRST transformation. The deformation of the BRST charge is:
where and are the left and right SUSY transformations.
The corresponding unintegrated vertex is:
What happens is where is a conserved current.
Flat space superstring has a ghost number 1 conserved charge anticommuting with
(and different from ).
We are deforming the BRST transformation without deforming the action.
But what happens if we deform like this:
(different sign) This actually satisfies all the axioms we introduced until now. But this deformation
does not correspond to anything in SUGRA. It is unphysical.