0
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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 $H \rightarrow H$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.