Counting Supertubes

TL;DR

This work tackles counting the quantum states of a D0-F1 supertube by directly quantizing the linearized D2-Born-Infeld action about a round tube. By gauge fixing and expanding to quadratic order, it shows the BPS sector forms a 1+1 right-moving CFT with central charge $c=12$ and uses the Cardy formula to obtain the entropy $S = 2\pi \sqrt{2 (Q_{D0} Q_{F1} - J)}$, confirming that supertubes capture the degeneracy of generic D0-F1 bound states. The analysis yields a discrete spectrum despite infinite zero-frequency modes, clarifying the marginal-bound-state nature of the configurations. The results connect to broader discussions of three-charge black holes and microstate geometries, outlining pathways to extend the method to more complex bound states while highlighting the geometric labeling of microstates provided by the supertube description.

Abstract

The quantum states of the supertube are counted by directly quantizing the linearized Born-Infeld action near the round tube. The result is an entropy $S = 2π\sqrt{2 (Q_{D0}Q_{F1}-J)}$, in accord with conjectures in the literature. As a result, supertubes may be the generic D0-F1 bound state. Our approach also shows directly that supertubes are marginal bound states with a discrete spectrum. We also discuss the relation to recent suggestions of Mathur et al involving three-charge black holes.