On a critical dimension in a spherical black brane phase transition

We study the Gregory-Laflamme instability of a large uniform black brane wrapping a 2-sphere compactification manifold. This paper continues the work of Kol and Sorkin (2006 Class. Quantum Grav. 23 4563), where the compactifications on p-torus were considered. The new features of the spherical case are the non-zero curvature of the compactification manifold and the absence of the rescaling symmetry due to a built-in stabilization mechanism. We calculate the order of the phase transition in dependence on the number d of extended dimensions using the Landau-Ginzburg approach. It is found that for d > 11, a uniform spherical black brane in a microcanonical ensemble exhibits a smooth second-order phase transition towards a stable branch of non-uniform black brane solutions. The critical number of extended dimensions, for which there is a change in the order of the phase transition, is different for microcanonical and canonical ensembles and does not coincide with the critical number of dimensions in the case of the flat toric compactifications. We briefly discuss the origin of this mismatch in the orders of phase transition for the different ensembles.