Geometry of D1-D5-P bound states

TL;DR

This paper analyzes the geometric structure of supersymmetric D1-D5-P bound states in 6D supergravity, revealing a pseudo-hyperkahler base with signature change and characterizing 3-charge microstates via a hypersurface analogue of the 2-charge central curve.

Contribution

It decomposes known bound state geometries into base-fiber form and uncovers the pseudo-hyperkahler structure with signature change, advancing understanding of black hole microstates.

Findings

Base geometry is pseudo-hyperkahler with signature change.

3-charge geometries characterized by a hypersurface analogous to the 2-charge central curve.

Decomposition reveals geometric features of bound states in supergravity.

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 $S^1\times S^3$.