Cardy matches Bethe on the Surface: a Tale of a Brane and a Black Hole

We consider the insertion of a Gukov-Witten surface defect in $\mathrm{SU}(N)$ $\mathcal{N}=4$ SYM corresponding to a probe D3-brane in the holographic dual setup. The defect gives rise to a $4d$-$2d$ coupled system encoding the entropy of the dual perturbed black hole, which can be extracted from the corresponding Superconformal Index. Elaborating on previous studies, we refine the results using both a saddle-point and a Bethe-Ansatz approach. The consistency of our computation is corroborated by the complete agreement between the two results in the appropriate regime of fugacities. Eventually, the sub-leading structure, emerging from our analysis, provides a suggestive EFT interpretation for the addition of the defect to the $4d$ system, mirroring the behavior of the probe D3-brane in the gravity dual.