Reminder about the dilaton
Let us remember how the dilaton field appears in the bosonic string theory.
The worldsheet sigma-model of the bosonic string theory is very simple:
If we do the Weyl transformation
anomaly is proportional to:
One would think naively, that the condition of vanishing of this Weyl anomaly implies the vacuum Einstein equation:
But this is too strong. Sometimes even when
, we can cancel the anomaly by
adding a local counterterm
. Suppose that we can find a harmonic function
Then let us add to the action the term:
is the worldsheet curvature. Then the variation of this counterterm under the Weyl
transformation compensates the anomaly.