D6+D0 and Five Dimensional Spinning Black Hole

We study the system of D6+D0 branes at sub-stringy scale. We show that the proper description of the system, for large background field associated with the D0-branes, is via spinning chargeless black holes in five dimensions. The repulsive force between the D6-branes and the D0-branes is understood through the centrifugal barrier. We discuss the implication on the stability of the D6+D0 solution. As it is well know there is a repulsive force between D6-branes and D0branes both at short and at large distances [1]. On the other hand a solution corresponding to a background of D0-branes on D6-branes exist [2]. The background is stable classically, at least to quadratic order. Since super Yang-Mills (SYM) theory in seven dimensions is a non-renormalizable theory one cannot test the stability of the background at the quantum level. The situation would have been different if there had been an underline theory which flows at the IR to SYM. However, such a theory, which does not involve gravity, does not exist [3, 4, 5, 6]. In this short note we take advantage of the recent progress in the understanding of the relation between the near horizon geometry of a given branes configuration and the field theory living on the branes [7] and study the near horizon geometry of D6-branes with a constant field associated with the D0-brane. We consider the “decoupling” limit while keeping the super-Yang-Mills coupling constant and the field strength, associated with the D0-brane background, fixed. We find that the near horizon geometry is that of a spinning black hole in five dimensions. When the field strength is large (compared to g −4/3 YM ) the size of the black hole horizon is large (compared to the Planck scale) and hence the supergravity solution can be trusted in the analysis of the stability of the D0-branes background. Before we consider the D6+D0 system let us review the decoupling limit of D6-branes. The “decoupling” limit is defined as follows [8] U = r α′ = fixed, g YM = (2π) 4gsα ′3/2 = fixed, α ′ → 0. (1) In this limit we keep the field theory energies and coupling constant fixed while taking α ′ to zero. This suggest that the D6-branes decouple from the bulk. However, as was noticed in [3, 4] in this limit R11 ∝ 1/α ′ → ∞, which means that the right description of the system is in M-theory as an ALE space with AN−1 singularity (where N is the number of D6-branes). Note