Geometry of D1-D5-P bound states

TL;DR

The paper investigates true bound states of the D1-D5-P system by expressing known 3-charge BPS geometries as a base×fiber construction with a Gibbons-Hawking base. It exposes a pseudo-hyperkahler base whose signature flips across the f=0 surface, whose topology is an orbifolded S^1×S^3 in the near-horizon limit, and analyzes regularity by showing how fiber terms cancel base divergences to produce smooth 6D geometries. The work highlights how different f=0 surface shapes may classify 3-charge microstates, relates a bound-state surface area to entropy, and emphasizes the role of harmonic-function structures in constraining viable bound-state geometries. Overall, it provides a framework to identify and understand genuine D1-D5-P bound states within the broader space of supersymmetric 6D solutions and connects geometric features to microstate counting ideas.

Abstract

Supersymmetric solutions of 6-d supergravity (with two translation symmetries) can be written as a hyperkahler base times a 2-D fiber. The subset of these solutions which correspond to true bound states of D1-D5-P charges give microstates of the 3-charge extremal black hole. To understand the characteristics shared by the bound states we decompose known bound state geometries into base-fiber form. The axial symmetry of the solutions make the base Gibbons-Hawking. We find the base to be actually `pseudo-hyperkahler': The signature changes from (4,0) to (0,4) across a hypersurface. 2-charge D1-D5 geometries are characterized by a `central curve' $S^1$; the analogue for 3-charge appears to be a hypersurface that for our metrics is an orbifold of $S^1\times S^3$.