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:
| (5) |
If we do the Weyl transformation
, the
anomaly is proportional to:
| (6) |
One would think naively, that the condition of vanishing of this Weyl anomaly implies the vacuum Einstein equation:
| (7) |
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:
| (8) |
Then let us add to the action the term:
| (9) |
where
is the worldsheet curvature. Then the variation of this counterterm under the Weyl
transformation compensates the anomaly.