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Spectral Flow, and the Spectrum of Multi-Center Solutions

Iosif Bena, Nikolay Bobev, Nicholas P. Warner

TL;DR

The paper demonstrates that spectral flow, implemented as coordinate transformations mixing U(1) directions in six-dimensional supergravity, can map smooth multi-center bubbling geometries based on a Gibbons-Hawking Taub-NUT base to configurations containing two-charge supertubes and vice versa. By exploiting SL(2,Z) actions and generalized spectral flow, the authors show that the space of smooth three-charge microstate geometries is, classically, infinite-dimensional due to arbitrary supertube profiles, and they establish detailed regularity conditions that persist under these flows. They further analyze multi-center D6-D4-D2-D0 configurations, clarifying when such solutions represent bound states, and argue that many apparently bound four-dimensional configurations are unbound from a six-dimensional perspective. These results provide a powerful framework for generating large families of microstate geometries, relate different duality frames, and pave the way for entropy counting and holographic interpretations of black hole microstates.

Abstract

We discuss "spectral flow" coordinate transformations that take asymptotically four-dimensional solutions into other asymptotically four-dimensional solutions. We find that spectral flow can relate smooth three-charge solutions with a multi-center Taub-NUT base to solutions where one or several Taub-NUT centers are replaced by two-charge supertubes, and vice versa. We further show that multi-parameter spectral flows can map such Taub-NUT centers to more singular centers that are either D2-D0 or pure D0-brane sources. Since supertubes can depend on arbitrary functions, we establish that the moduli space of smooth horizonless black hole microstate solutions is classically of infinite dimension. We also use the physics of supertubes to argue that some multi-center solutions that appear to be bound states from a four-dimensional perspective are in fact not bound states when considered from a five- or six-dimensional perspective.

Spectral Flow, and the Spectrum of Multi-Center Solutions

TL;DR

The paper demonstrates that spectral flow, implemented as coordinate transformations mixing U(1) directions in six-dimensional supergravity, can map smooth multi-center bubbling geometries based on a Gibbons-Hawking Taub-NUT base to configurations containing two-charge supertubes and vice versa. By exploiting SL(2,Z) actions and generalized spectral flow, the authors show that the space of smooth three-charge microstate geometries is, classically, infinite-dimensional due to arbitrary supertube profiles, and they establish detailed regularity conditions that persist under these flows. They further analyze multi-center D6-D4-D2-D0 configurations, clarifying when such solutions represent bound states, and argue that many apparently bound four-dimensional configurations are unbound from a six-dimensional perspective. These results provide a powerful framework for generating large families of microstate geometries, relate different duality frames, and pave the way for entropy counting and holographic interpretations of black hole microstates.

Abstract

We discuss "spectral flow" coordinate transformations that take asymptotically four-dimensional solutions into other asymptotically four-dimensional solutions. We find that spectral flow can relate smooth three-charge solutions with a multi-center Taub-NUT base to solutions where one or several Taub-NUT centers are replaced by two-charge supertubes, and vice versa. We further show that multi-parameter spectral flows can map such Taub-NUT centers to more singular centers that are either D2-D0 or pure D0-brane sources. Since supertubes can depend on arbitrary functions, we establish that the moduli space of smooth horizonless black hole microstate solutions is classically of infinite dimension. We also use the physics of supertubes to argue that some multi-center solutions that appear to be bound states from a four-dimensional perspective are in fact not bound states when considered from a five- or six-dimensional perspective.

Paper Structure

This paper contains 18 sections, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Three charge solutions
  3. Five-dimensional BPS solutions
  4. Six-dimensional BPS solutions and dimensional reduction
  5. Spectral Flow
  6. Solutions with a Gibbons-Hawking base
  7. Review of solutions with a Gibbons-Hawking bases
  8. Spectral flow in Gibbons-Hawking metrics
  9. $SL(2,\mathbb{Z})$ transformations of bubbling solutions
  10. Regularity and the Bubble Equations
  11. Two-Charge Supertubes and Spectral Flow
  12. Bubbling Geometries from Supertubes - One Supertube in Taub-NUT
  13. Bubbling Geometries from Supertubes - Many Supertubes in Taub-NUT
  14. Supertubes from Bubbling Geometries
  15. Generalized Spectral Flow
  16. ...and 3 more sections

Figures (1)

  • Figure 1: The "hedgehog" quiver corresponding to the unbound multi-center solution generated from the spectral flow of many concentric supertubes.