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 , the 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 such that:


Then let us add to the action the term:


where is the worldsheet curvature. Then the variation of this counterterm under the Weyl transformation compensates the anomaly.