String gravity in $D=4$

We revisit the four-dimensional theory of gravity that arises from string theory with higher-derivative corrections. By compactifying and truncating the ten-dimensional effective action of heterotic string theory at first order in $\alpha'$, and carefully dealing with field redefinitions, we show that the four-dimensional theory takes the form of an axidilaton model where the scalars couple to the Gauss-Bonnet and Pontryagin densities. Thus, the actual string gravity is a generalization of the well-studied Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons models. Using this action we compute the stringy corrections to the Kerr geometry and we obtain, for the first time, the corrections to the entropy of the Kerr black hole at order $\alpha'^2$. We check that the first law of black hole mechanics is satisfied and discuss several properties of the solution. Our results suggest that there exist black hole solutions with $J>M^2$ and therefore the extremal ratio $J/M^2$ must be modified positively.