Removal of nonphysical states
We have seen that there are some unphysical deformations. One way to look at them is to consider the so-called beta deformation which is adding to the action the wedge product of two global symmetry currents:

where

(5)

The corresponding unintegrated vertex is . This is nonphysical when where is the structure constants of the global symmetry group (super-Poincare). Notice that:

Therefore, we must postulate that:


But what happens if we, still, deform the action with (5) with ? The worldsheet action will be perfectly consistent as a classical action, and enjoying a nice BRST symmetry. At the classical level, nothing is wrong. But:
  • the quantum theory will be anomalous at the level of one loop

(anomaly in the holomorphicity of the and antiholomorphicity of )
  • Still, why SUGRA equations are not satisfied when ?

Because the equation for the dilaton becomes inconsistent.