On $\alpha'$-effects from $D$-branes in $4d \; \mathcal{N} = 1$

In this work we study type IIB Calabi-Yau orientifold compactifications in the presence of space-time filling D7-branes and O7-planes. In particular, we conclude that $\alpha'^2 g_s$-corrections to their DBI actions lead to a modification of the four-dimensional $\mathcal{N}=1$ K\"ahler potential and coordinates. We focus on the one-modulus case of the geometric background i.e. $h^{1,1}=1$ where we find that the $\alpha'^2 g_s$-correction is of topological nature. It depends on the first Chern form of the four-cycle of the Calabi-Yau orientifold which is wrapped by the D7-branes and O7-plane. This is in agreement with our previous F-theory analysis and provides further evidence for a potential breaking of the no-scale structure at order $\alpha'^2 g_s$. Corrected background solutions for the dilaton, the warp-factor as well as the internal space metric are derived. Additionally, we briefly discuss $\alpha'$-corrections from other D$p$-branes.