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Branes in the Euclidean AdS_3

B. Ponsot, V. Schomerus, J. Teschner

TL;DR

This work delivers an exact BCFT description of maximally symmetric branes in the Euclidean AdS$_3$ background, focusing on AdS$_2$-type branes localized along Euclidean ${ m AdS}_2\subset{ m AdS}_3$ and a second SU(2)-symmetric class of spherical branes. It constructs precise one-point functions and boundary states, derives open-string reflection amplitudes, and computes open-string spectral densities, validating them through Cardy-type world-sheet duality and a detailed factorization analysis involving degenerate fields. The results reveal how closed-string couplings encode open-string spectra and how consistent boundary data emerges from gluing conditions and degenerate-field factorization, with intricate special functions ($\Gamma_k$, $S_k$, Harish-Chandra $c$-functions) organizing the stringy corrections. The findings have implications for D-branes in 2D black-hole geometries via the $H_3^+/\mathbb{R}_\tau$ coset and provide a framework for extending the Cardy program to non-rational current-algebra models, including potential higher-point boundary correlators and holographic interpretations.

Abstract

In this work we propose an exact microscopic description of maximally symmetric branes in a Euclidean $AdS_3$ background. As shown by Bachas and Petropoulos, the most important such branes are localized along a Euclidean $AdS_2 \subset AdS_3$. We provide explicit formulas for the coupling of closed strings to such branes (boundary states) and for the spectral density of open strings. The latter is computed in two different ways first in terms of the open string reflection amplitude and then also from the boundary states by world-sheet duality. This gives rise to an important Cardy type consistency check. All the results are compared in detail with the geometrical picture. We also discuss a second class of branes with spherical symmetry and finally comment on some implications for D-branes in a 2D back hole geometry.

Branes in the Euclidean AdS_3

TL;DR

This work delivers an exact BCFT description of maximally symmetric branes in the Euclidean AdS background, focusing on AdS-type branes localized along Euclidean and a second SU(2)-symmetric class of spherical branes. It constructs precise one-point functions and boundary states, derives open-string reflection amplitudes, and computes open-string spectral densities, validating them through Cardy-type world-sheet duality and a detailed factorization analysis involving degenerate fields. The results reveal how closed-string couplings encode open-string spectra and how consistent boundary data emerges from gluing conditions and degenerate-field factorization, with intricate special functions (, , Harish-Chandra -functions) organizing the stringy corrections. The findings have implications for D-branes in 2D black-hole geometries via the coset and provide a framework for extending the Cardy program to non-rational current-algebra models, including potential higher-point boundary correlators and holographic interpretations.

Abstract

In this work we propose an exact microscopic description of maximally symmetric branes in a Euclidean background. As shown by Bachas and Petropoulos, the most important such branes are localized along a Euclidean . We provide explicit formulas for the coupling of closed strings to such branes (boundary states) and for the spectral density of open strings. The latter is computed in two different ways first in terms of the open string reflection amplitude and then also from the boundary states by world-sheet duality. This gives rise to an important Cardy type consistency check. All the results are compared in detail with the geometrical picture. We also discuss a second class of branes with spherical symmetry and finally comment on some implications for D-branes in a 2D back hole geometry.

Paper Structure

This paper contains 56 sections, 4 theorems, 229 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Strings on $H_3^+$ -- the semi-classical limit
  3. Bulk geometry and the closed string action
  4. Geometry of $H_3^+$.
  5. The string action.
  6. The currents.
  7. Brane geometry and the boundary conditions
  8. General results.
  9. The $AdS_2$ branes.
  10. Spherical branes.
  11. Semi-classical limit of closed string couplings
  12. Harmonic analysis on $H_3^+$.
  13. The $AdS_2$-branes.
  14. The spherical branes.
  15. Semi-classical limit of open string spectra
  16. ...and 41 more sections

Key Result

Proposition A.1

The distributions $D^j_{\epsilon}$ can be represented in the following simple form Here, $h$ is an element of $H_3^+$ which is parametrized by $(\psi, \chi,\nu)$ that we introduced in equation (param1).

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Claim 1
  • Claim 2
  • Remark 5
  • Remark 6
  • Remark 7
  • Proposition A.1
  • ...and 4 more