$\alpha'$-corrected black holes in String Theory

We consider the well-known solution of the Heterotic Superstring effective action to zeroth order in $\alpha'$ that describes the intersection of a fundamental string with momentum and a solitonic 5-brane and which gives a 3-charge, static, extremal, supersymmetric black hole in 5 dimensions upon dimensional reduction on $\mathrm{T}^{5}$. We compute explicitly the first-order in $\alpha'$ corrections to this solution, including $\mathrm{SU}(2)$ Yang-Mills fields which can be used to cancel some of these corrections and we study the main properties of this $\alpha'$-corrected solution: supersymmetry, values of the near-horizon and asymptotic charges, behavior under $\alpha'$-corrected T-duality, value of the entropy (using Wald formula directly in 10 dimensions), existence of small black holes etc. The value obtained for the entropy agrees, within the limits of approximation, with that obtained by microscopic methods. The $\alpha'$ corrections coming from Wald's formula prove crucial for this result.