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In Superstring Theory vol.1 by Green, Schwarz and Witten (GSW) the low energy effective action of the bosonic string is given by:

$$S=-\frac{1}{2k_0^2}\int d^{26}x\sqrt{-g}e^{-2\Phi}({\cal R}-4\partial_\mu\Phi\partial^\mu\Phi+\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}).\tag{GSW3.4.57}$$

But in String Theory vol.1 by Polchinski and in David Tong's online lectures on string theory the low energy effective action of the bosonic string is given by:

$$S=\frac{1}{2k_0^2}\int d^{26}x\sqrt{-g}e^{-2\Phi}({\cal R}+4\partial_\mu\Phi\partial^\mu\Phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}). \tag{P3.7.20/T7.16}$$

As far as I can tell they are using the same conventions so why the difference?

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    $\begingroup$ The real question is: are the $\beta$ functions the same in GSW and Polchinski. In Polchinski the effective action is given and it is stated that these lead to the $\beta$ function equations. I have checked this calculation in detail and have found it to be correct. I have the detailed calculation available, if you want it. $\endgroup$ Apr 27, 2021 at 16:42
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    $\begingroup$ Ok thanks I accept the Polchinski action leads to his $\beta$ functions. I would need to check if GSW $\beta$ functions are same as Polchinski. $\endgroup$ Apr 27, 2021 at 19:45

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Since all the mentioned authors use the same Minkowskian signature $(-,+,\ldots, +)$, it seems the difference is caused by different sign convention for the scalar curvature.

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