Localised $\mathrm{AdS}_3\times \mathrm{S}^3\times \mathbb{T}^4$ Black Holes

We numerically construct asymptotically global $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ black holes in type IIB supergravity, with $\mathbb{R}_t \times SO(2) \times SO(3) \times U(1)^4$ symmetry, localised on the $\mathrm{S}^3$ and translationally invariant along the torus. These solutions, with horizons whose spatial cross sections have $\mathrm{S}^4 \times \mathbb{T}^4$ topology, dominate the microcanonical ensemble at low energies. At higher energies, a first-order phase transition occurs to $\mathrm{BTZ} \times \mathrm{S}^3 \times \mathbb{T}^4$ black holes $-$ possessing $\mathbb{R}_t \times SO(2) \times SO(4) \times U(1)^4$ symmetry and $\mathrm{S}^1 \times \mathrm{S}^3 \times \mathbb{T}^4$ horizon topology. By the AdS/CFT correspondence, this transition reflects the spontaneous breaking of the $SO(4)$ R-symmetry of the D1-D5 CFT$_2$ to $SO(3)$. We also compute the expectation value of the scalar operator with lowest conformal dimension in the low-energy phase. Our $SO(3)$-localised black holes $-$ together with the $U(1)^2$-localised solutions of [1] $-$ point to a rich landscape of novel black holes that may approach the $\mathrm{CFT}_2$ sparseness bootstrap condition, and shed light on how macroscopic entanglement in thermal phases encodes microscopic structure via internal directions.