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).
