Sign Wrong in Effective Action of Bosonic String?

In David Tong's Lectures in String Theory Chapter 7 he sketches a derivation of the low-energy effective action of the bosonic string $$(7.16)$$: $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\mathcal{R}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_\mu\Phi\partial^\mu\Phi\Big).\tag{1}$$ Towards the bottom of page $$162$$ he states that the stress-energy tensor $$T_{\alpha\beta}$$ is given by $$T_{\alpha\beta}=\frac{4\pi}{\sqrt{g}}\frac{\delta S}{\delta g^{\alpha\beta}}.\tag{2}$$ But normally the stress-energy tensor is defined by the negative of the expression in Eqn $$(2)$$. For example see the expression for $$T_{\mu\nu}$$ in the Derivation of Einstein field equations section of the Einstein-Hilbert Action Wiki page: $$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_M)}{\delta g^{\mu\nu}}\tag{3}.$$ If we use an expression like Eqn $$(2)$$, without the minus sign, as a definition of $$T_{\mu\nu}$$ rather than Eqn $$(3)$$ then the Einstein-Hilbert action has to change to: $$S=\int\Big[-\frac{1}{2\kappa}\mathcal{R}+\mathcal{L}_M\Big]\sqrt{-g}\ d^4x.$$ Thus to be consistent with the definition of $$T_{\alpha\beta}$$ in Eqn$$(2)$$ the low-energy effective action should be given by: $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\mathcal{-R}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_\mu\Phi\partial^\mu\Phi\Big).\tag{4}$$ Has David Tong got the sign of the $$\mathcal{R}$$ term wrong in his expression for the low-energy effective action of the bosonic string?

If he has got it wrong then so have the authors of Sigma Models and String Theory, which David Tong cites, as they also use a definition of $$T_{\alpha\beta}$$ without a minus sign (2.3) to derive a bosonic string effective action with a positive $$\mathcal{R}$$ term (3.56).

• You are mixing up Euclidean and Lorentzian signatures. You get extra minus signs when you go from one signature to the other via a Wick rotation. Oct 13, 2021 at 12:56
• @Prahar Please convert that to an answer, preferably with more detail.
– J.G.
Oct 13, 2021 at 13:08
• The Polyakov action in (1.22)/(1.30) has an overall minus sign, therefore $T_{ab}$ above (1.30) or in (4.4) has a minus sign, so that (1.32) looks nice. In (7.1) however the action $S$ has an overall $+$ sign, that's why $T_{ab}$ on p. 162 gets changed to a plus sign. There are comments around (1.30) and (4.4) on how the overall coefficients are just choices, e.g. why does your (2) getting a factor of $2 \pi$ compared to your (3) not cause any concerns. Oct 13, 2021 at 13:39
• Also: see around (4.4), where one uses the Euclidean metric (e.g. immediately below (4.4), yet he still has a minus sign in (4.4), so at least in this context it's not about the signature it's just about the overall choice of sign in the actions in (1.30) vs. (7.1). Oct 13, 2021 at 13:50
• Possible duplicate by OP: physics.stackexchange.com/q/632350 Oct 13, 2021 at 16:41

1. Eq. (1) on p. 168 is apparently in Minkowskian signature because of the minus under the square root. On p. 9 Tong explains it is $$(-,+,\ldots,+)$$ sign convention.
2. Eq. (2) on p. 162 is apparently in Euclidean signature because of the Kronecker delta $$\delta_{\alpha\beta}$$ in the first formula on p. 162.
3. Eq. (3) on the Wikipedia page is in Minkowskian signature $$(-,+,+,+)$$.