Invariance of the low energy effective string action

Question

It is well known that the action of General Relativity $$S = \frac{1}{16\pi G}\int R\;\sqrt{-g} d^D X$$ is invariant under "diffeomorphisms".

The low energy effective action for bosonic strings is

$$S = \frac{1}{2\kappa_0^2}\int d^D X\; \sqrt{-g}\; \mathrm{e}^{-2\Phi}\,(R-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4 \partial_{\mu}\Phi\partial^{\mu}\Phi). \; \; (H=dB)$$ Is also the low energy effective action invariant under "diffeomorphism"?

Perhaps there is a sort of generalization to include gauge invariance in $B$ (Kalb-Ramond field) and in the dilaton. please, give references.

Answer 1

The best reference is Polchinski's textbook (vol.'s 1 and 2).

This low-energy action is indeed invariant under diffeomorphisms--all objects appearing in the integrand are geometric invariants. This means that there are no un-contracted indices and also that they transform as scalars under diffeomorphisms. And the last ingredient is present too--the $\sqrt{-g}$ volume form. You can directly verify that under a change of coordinates the above action is invariant.

Also also the kinetic term for the Kalb-Ramond field is gauge invariant in exactly the same way the kinetic term for usual $U(1)$ electromagnetism is variant. In the language of forms, $B \rightarrow B + dA$, and so $H \rightarrow H$.