A Geometric Look on the Microstates of Supertubes

We give a geometric interpretation of the entropy of the supertubes with fixed conserved charges and angular momenta in two different approaches using the DBI action and the supermembrane theory. By counting the geometrically allowed microstates, it is shown that both the methods give consistent result on the entropy. In doing so, we make the connection to the gravity microstates clear. 1 Introduction Supertubes are tubular shaped bound states of D0-branes, fundamental strings (F1) and D2-branes [1]-[28]. While having translational invariance in the axial direction along which the F1 strings are stretched, the cross sectional shape of the supertubes may be arbitrary in the eight transverse dimensions. As shown in Ref. [6], the cross sectional shape could be either open and stretched to infinity or closed but here we would like to focus on the closed cases. Let us begin our discussion with the cases where the cross sectional curve lies in x 1 and x 2 plane. The supertube then carries an angular momentum density J = J 12 proportional to the cross sectional area. For the fixed conserved charges, the moduli space of supertubes is consisting of the geometric fluctuations of the cross sectional shape [26]. Since the angular momentum is fixed, the fluctuation of the curve has to be area preserving. The length L of the cross sectional curve is further limited by √ Q 0 Q 1 where Q 0 and Q 1 denote lineal D0 density in the axis direction and F1 charges divided by 2π respectively. Thus one has the restriction of the length by J/T 2 ≤ L ≤ Q 0 Q 1 /T 2 , where T 2 is the D2-brane tension. This space of arbitrary fluctuation of the curve forms an infinite dimensional moduli space. For given curve, the magnetic field representing the density of D0 may be arbitrary with total number of D0-branes fixed. Moreover the shape may fluctuate into the six more transverse directions. Hence eight arbitrary bosonic functional fluctuations are involved as the moduli deformation. Since the supertubes involve a nonvanishing electric field and linear momentum densities fixed by the shape of the curve, the above moduli space is not a configuration space but a phase space. The supertubes allow corresponding supergravity solutions [4, 9] of an arbitrary cross sectional shape and arbitrary density of D0-brane as a function of the world-volume coordinate φ of the curve direction. …