Critical lumpy black holes in AdS${}_p\times S^q$

In this paper we study lumpy black holes with AdS${}_p \times S^q$ asymptotics, where the isometry group coming from the sphere factor is broken down to SO($q$). Depending on the values of $p$ and $q$, these are solutions to a certain Supergravity theory with a particular gauge field. We have considered the values $(p,q) = (5,5)$ and $(p,q) = (4,7)$, corresponding to type IIB supergravity in ten dimensions and eleven-dimensional supergravity respectively. These theories presumably contain an infinite spectrum of families of lumpy black holes, labeled by a harmonic number $\ell$, whose endpoints in solution space merge with another type of black holes with different horizon topology. We have numerically constructed the first four families of lumpy solutions, corresponding to $\ell = 1, 2^+, 2^-$ and $3$. We show that the geometry of the horizon near the merger is well-described by a cone over a triple product of spheres, thus extending Kol's local model to the present asymptotics. Interestingly, the presence of non-trivial fluxes in the internal sphere implies that the cone is no longer Ricci flat. This conical manifold accounts for the geometry and the behavior of the physical quantities of the solutions sufficiently close to the critical point. Additionally, we show that the vacuum expectation values of the dual scalar operators approach their critical values with a power law whose exponents are dictated by the local cone geometry in the bulk.