Beyond the near-horizon limit: Stringy corrections to Heterotic Black Holes

We study the first-order in $\alpha'$ corrections to 4-charge black holes (with the Reissner-Nordstr\"om black hole as a particular example) beyond the near-horizon limit in the Heterotic Superstring effective action framework. The higher-curvature terms behave as delocalized sources in the equations of motion and in the Bianchi identity of the 3-form. For some charges, this introduces a shift between their values measured at the horizon and asymptotically. Some of these corrections and their associated charge shifts, but not all of them, can be canceled using appropriate SU$(2)$ instantons for the heterotic gauge fields. The entropy, computed using Wald's formula, is in agreement with the result obtained via microstate counting when the delocalized sources are properly taken into account.