Spectral Flow of the Non-Supersymmetric Microstates of the D1-D5-KK System

TL;DR

This work extends the notion of spectral flow to non-supersymmetric microstates of the D1-D5-KK system by implementing a coordinate transformation that adds a momentum charge, producing D1-D5-P-KKM solutions from smooth D1-D5-KK geometries. The construction preserves regularity and, in the near-core AdS$_3 imes$ S$^3$ regime, aligns with the familiar spectral-flow picture in the dual CFT and with known three- and four-charge cores (JMRT05). A careful quantization of the flow parameter ensures the identifications remain consistent, and the near-core limit reproduces the expected AdS$_3 imes$ S$^3$ structure, reinforcing the fuzzball program's geometric realization for non-supersymmetric states. The results demonstrate a concrete link between four-dimensional asymptotics and four-charge microstate geometries, while highlighting remaining challenges in constructing more general non-supersymmetric smooth solutions.

Abstract

We show that a realisation of spectral flow as a coordinate transformation for asymptotically four-dimensional solutions can be extended to the non-supersymmetric case. We apply this transformation to smooth geometries describing microstates of the D1-D5-KK monopole system in type IIB supergravity compactified on a six-torus, and obtain solutions with an additional momentum charge. We study the supersymmetric and near-core limits of this construction.