# Invariance of the low energy effective string action

+ 1 like - 0 dislike
85 views

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.

This post imported from StackExchange Physics at 2015-02-10 11:03 (UTC), posted by SE-user Anthonny

retagged Feb 10, 2015

+ 1 like - 0 dislike

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 $H \rightarrow H$.

This post imported from StackExchange Physics at 2015-02-10 11:03 (UTC), posted by SE-user Surgical Commander
answered Feb 10, 2015 by (155 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.