A Geometric Look on the Microstates of Supertubes

TL;DR

This work develops two geometric frameworks to count the microstates of supertubes with fixed charges $(Q_0,Q_1)$ and angular momenta: (i) a DBI description of a D2-brane with electric and magnetic fluxes, including bosonic and fermionic flat directions, and (ii) an 11-dimensional M-theory/supermembrane picture. In the DBI approach, small cross-sectional fluctuations around a circular tube are quantized on a phase-space of bosonic and fermionic moduli, yielding entropy scales such as $S \,=\,\pi\sqrt{\tfrac{4\pi}{3}\,L_z\,(Q_0 Q_1-J)}$ for the near-circular bosonic counting, and the full near-circular (bosonic+fermionic) counting gives $S=4\pi\sqrt{\pi L_z\,(Q_0 Q_1-J)}$, with a multi-Cartan generalization to $J_a$ giving $S=4\pi\sqrt{\pi L_z\,(Q_0 Q_1-\sum_a J_a)}$. The M-theory analysis reproduces the same leading degeneracy, yielding $S\sim 4\pi\sqrt{\pi L_z\,(Q_0 Q_1-\sum_a J_a)}$, thereby validating the gravity microstate interpretation via geometric phase-space microstates. The results delineate a regime where both descriptions overlap (decoupling/open-string limit) and connect the worldvolume moduli to gravity microstates, offering a concrete bridge between geometric fluctuations and microstate counting in string/M-theory.

Abstract

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.