Bosonic string analogy

We have constructed the nontrivial deformation of the BRST transformation, which does not require the deformation of the action.

There is a similar phenomenon in the bosonic string theory. Let us consider the bosonic string, with the worldsheet action:

Linear dilaton background corresponds to the deformation:

where is the dilaton gradient, and is the worldsheet curvature.

On the flat worldsheet, the action is not deformed. However, the BRST charge is deformed, because the energy-momentum tensor is deformed:

(the so-called Feigin-Fuks theory). Lightlike is needed to keep the central charge 26 (i.e. to avoid backreaction).