On the existence of scaling multi-centered black holes

TL;DR

This paper establishes necessary conditions on charges for multi-centered black hole solutions in supergravity, revealing their relation to quiver representations and the Coulomb-Higgs duality.

Contribution

It generalizes the conditions for the existence of scaling multi-centered black holes to any number of centers and links these conditions to quiver representation stability.

Findings

Necessary charge conditions for scaling solutions.

Conditions are also necessary at the attractor point.

Link between black hole solutions and quiver representations.

Abstract

For suitable charges of the constituents, the phase space of multi-centered BPS black holes in N=2 four-dimensional supergravity famously exhibits scaling regions where the distances between the centers can be made arbitrarily small, so that the bound state becomes indistinguishable from a single-centered black hole. In this note we establish necessary conditions on the Dirac product of charges for the existence of such regions for any number of centers, generalizing the standard triangular inequalities in the three-center case. Interestingly, the same conditions are necessary for the existence of multi-centered solutions at the attractor point. We prove that similar conditions are also necessary for the existence of self-stable representations of the corresponding quiver, as suggested by the duality between the Coulomb and Higgs branches of supersymmetric quantum mechanics.