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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?

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  • $\begingroup$ is metric signature mostly minus? $\endgroup$
    – Kosm
    Jun 7, 2020 at 20:55
  • $\begingroup$ Polchinski is mostly positive $\endgroup$ Jun 7, 2020 at 20:57
  • $\begingroup$ 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 $-4\partial\sigma\partial\sigma$. $\endgroup$
    – Kosm
    Jun 7, 2020 at 21:02
  • $\begingroup$ @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. $\endgroup$
    – secavara
    Jun 7, 2020 at 21:11
  • $\begingroup$ @secavara I see now, I missed the dilaton factor. $\endgroup$
    – Kosm
    Jun 7, 2020 at 21:14

1 Answer 1

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If you will excuse my quick scribbles (using ipad at the moment), here is something to try, let me know if it worked.

enter image description here

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  • $\begingroup$ Thanks a lot. How could I miss that? $\endgroup$ Jun 8, 2020 at 9:07

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