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.