# Graviton-Dilaton Action for Kaluza-Klein in Polchinski (8.1.9)

Polchinski uses the graviton-dilaton action (8.1.9) in his String Theory book $$S_1= \frac{1}{2\kappa_0^2}\int d^D x\, \sqrt{-G} e^{-2\Phi} \left[ {R} + 4 \nabla_\mu\Phi \nabla^\mu \Phi \right] \tag{8.1.9}$$ for the Kaluza-Klein theory. He then rewrites this in terms of the Kaluza-Klein fields as $$S_1= \frac{\pi R }{\kappa_0^2}\int d^d x\, \sqrt{-G_d} e^{-2\Phi+\sigma } \left[ {R_d} -4\partial_\mu\Phi \partial\sigma + 4 \partial_\mu\Phi \partial^\mu \Phi-\frac{1}{4} e^{2\sigma}F_{\mu\nu}F^{\mu\nu} \right]$$ I understand where most of this comes. The integration over $$x^d$$ gives $$2\pi R$$. We have $$\sqrt{-G}= e^\sigma \sqrt{-G_d}$$. We use (8.1.8), i.e. $$R = R_d -2e^{-\sigma} \nabla^2 e^\sigma-\frac{1}{4} e^{2\sigma} F_{\mu\nu}F^{\mu\nu}$$ and we replace $$\nabla_\mu \Phi$$ by $$\partial_\mu \Phi$$ as $$\Phi$$ is a space-time scalar (I assume this is correct?).

But how does the $$-2e^{-\sigma} \nabla^2 e^\sigma$$ change into $$-4\partial_\mu\Phi \partial\sigma$$? What am I missing?

• is metric signature mostly minus?
– Kosm
Jun 7, 2020 at 20:55
• Polchinski is mostly positive Jun 7, 2020 at 20:57
• then scalar kinetic terms should have minus sign, namely the $\Phi$ kinetic terms. Misprint perhaps. Also I think the $-4\partial\Phi\partial\sigma$ should be $-4\partial\sigma\partial\sigma$.
– Kosm
Jun 7, 2020 at 21:02
• @Kosm they provide a justification for the apparent wrong sign of the kinetic term in the paragraph after the equation. The mixed term seems to be legitimate but I have not found a justification. Jun 7, 2020 at 21:11
• @secavara I see now, I missed the dilaton factor.
– Kosm
Jun 7, 2020 at 21:14 