Thermodynamics of black holes with probe D-branes

Understanding how the thermodynamic properties of a black hole are modified when probed by D-branes is an important problem in AdS/CFT. This work focuses on a recently proposed black hole/D3-brane system in AdS₅×S⁵, which is dual to four-dimensional $\mathcal{N}$ \( \mathcal{N} \) = 4 SYM in the presence of a two-dimensional surface defect. The Laplace transform that extracts the asymptotic growth of states in this defect CFT naturally defines a thermodynamic approach in the gravitational side of the duality for which charges and entropy are real. Studying the superconformal defect index in a large-charge expansion for all values of N, we compute the leading correction to the entropy of the combined system, which matches precisely with its gravity counterpart.