# String frame and Einstein frame for a Dp-brane

The low energy effective action for $N$ D$p$-branes in the string frame is

$$S_\text{eff} = \frac{1}{16\pi G_{10}}\int d^{10}x \sqrt{-g}\ e^{-2\phi}(R+ 4(\partial\phi)^2+\cdots)$$

where $R$ is the Ricci curvature of $g$ and $\phi$ is the dilaton. There are another terms related to a antisymmetric tensor, but I'm not interested in them.

This gives us the metric:

$$ds^2 = H^{-1/2} (-dt^2 + dx^2_p) + H^{1/2} (dr^2 + r^2d\Omega_{8-p}^2)$$

and the dilaton

$$e^{-2\phi} = H^{\frac{p-3}{2}}$$

with

$$H(r) = 1 + \left(\frac{R}{r}\right)^{7-p} .$$

If I want to obtain the Einstein metric I need to do a proper Weyl transformation, in that case

$$g_{\mu\nu}\mapsto g^E_{\mu\nu} = e^{\phi/2}g_{\mu\nu}$$

My question is, in this tranformation do I have to put $H^{\frac{p-3}{8}}$ (the dilaton that I wrote above) or do I need to compute a new dilaton with the new action?

• Even if you will compute a new dialton with the new action why would it be different than the previous one i mean you only add "a phase"(I think at least) – Avrham Aton Sep 14 '14 at 16:32
• Would you be so kind and tell me what the source of your expressions is? The answer might depend on the definition of $\phi$. – Frederic Brünner Sep 14 '14 at 23:40
• @FredericBrünner Yes, it is the expression 39 in this PDF: members.ift.uam-csic.es/jfbarbon/Teaching_files/strbasics.pdf – dpravos Sep 15 '14 at 9:36
• At the end of the page $53$, you see that the action in the Einstein frame, and the dilaton field is the same that in equation $(39)$. The metrics, in the Eistein frame, in $10$ dimensions is simply $ds_E^2 = e^{\phi/2} ds^2$ – Trimok Sep 15 '14 at 9:49

No, the expression for the dilaton it's the same, being a scalar. In a generic dimension D the relation is:

$$g_{\mu\nu}^E = e^{-\frac{4\phi}{D-2}} g_{\mu\nu}^S$$

Having used:

$$L_{Einstein} = \sqrt{-g_E} R_E$$

$$L_{String} = \sqrt{-g_S} R_S e^{-2\phi}$$

References:

Basic Concepts In String Theory, B-L-T, pag. 700

Gravity and Strings, Ortin, pag. 414