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Branes in the 2D black hole

Sylvain Ribault, Volker Schomerus

TL;DR

This work provides a comprehensive boundary conformal field theory treatment of branes in the Euclidean 2D black hole (the cigar), deriving exact boundary states and open-string spectra for D0, D1, and D2 branes. It leverages a descent from the H3 model and connections to sine-Liouville theory to illuminate how discrete tip-localized closed-string modes modify boundary data, and it verifies consistency via Cardy-type dualities in annulus amplitudes. The results offer semi-classical pictures for each brane family, clarify the role of world-volume flux on D2 branes, and illuminate the near-horizon NS5-brane geometry with potential links to dual matrix models and little string theory. The paper also outlines key open issues, including boundary three-point functions and dynamical processes such as brane bound states, inviting further exploration of brane dynamics in this tractable setting.

Abstract

We present a comprehensive analysis of branes in the Euclidean 2D black hole (cigar). In particular, exact boundary states and annulus amplitudes are provided for D0-branes which are localized at the tip of the cigar as well as for two families of extended D1 and D2-branes. Our results are based on closely related studies for the Euclidean AdS3 model and, as predicted by the conjectured duality between the 2D black hole and the sine-Liouville model, they share many features with branes in Liouville theory. New features arise here due to the presence of closed string modes which are localized near the tip of the cigar. The paper concludes with some remarks on possible applications.

Branes in the 2D black hole

TL;DR

This work provides a comprehensive boundary conformal field theory treatment of branes in the Euclidean 2D black hole (the cigar), deriving exact boundary states and open-string spectra for D0, D1, and D2 branes. It leverages a descent from the H3 model and connections to sine-Liouville theory to illuminate how discrete tip-localized closed-string modes modify boundary data, and it verifies consistency via Cardy-type dualities in annulus amplitudes. The results offer semi-classical pictures for each brane family, clarify the role of world-volume flux on D2 branes, and illuminate the near-horizon NS5-brane geometry with potential links to dual matrix models and little string theory. The paper also outlines key open issues, including boundary three-point functions and dynamical processes such as brane bound states, inviting further exploration of brane dynamics in this tractable setting.

Abstract

We present a comprehensive analysis of branes in the Euclidean 2D black hole (cigar). In particular, exact boundary states and annulus amplitudes are provided for D0-branes which are localized at the tip of the cigar as well as for two families of extended D1 and D2-branes. Our results are based on closely related studies for the Euclidean AdS3 model and, as predicted by the conjectured duality between the 2D black hole and the sine-Liouville model, they share many features with branes in Liouville theory. New features arise here due to the presence of closed string modes which are localized near the tip of the cigar. The paper concludes with some remarks on possible applications.

Paper Structure

This paper contains 23 sections, 97 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Closed strings on the cigar
  3. The minisuperspace model
  4. The spectrum of the bulk CFT
  5. Coset construction from $H_3$
  6. The D0-branes
  7. Semi-classical description
  8. The exact solution
  9. D0-branes from descent
  10. Cardy consistency condition
  11. The D1-branes
  12. Semi-classical description
  13. The exact solution
  14. D1-branes from descent
  15. Cardy consistency condition
  16. ...and 8 more sections

Figures (8)

  • Figure 1: The 2d black hole
  • Figure 2: The cigar from $H_3$
  • Figure 3: A D1-brane in the cigar in $\rho$ and $u$ coordinates
  • Figure 4: A D1-brane in the cigar with open strings on it
  • Figure 5: A D1-brane in the cigar descending from an $AdS_2$ brane in $H_3$
  • ...and 3 more figures