Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Open Strings on AdS_2 Branes

Peter Lee, Hirosi Ooguri, Jongwon Park, Jonathan Tannenhauser

TL;DR

This paper analyzes open strings ending on AdS_2 branes within AdS_3 in an NS-NS background using the SL(2,R) WZW model. It shows that straight AdS_2 branes yield an open-string spectrum that is the holomorphic square root of the closed-string AdS_3 spectrum, with both short and long strings and full spectral flow, while curved AdS_2 branes break half the spectral flow and introduce ψ_0-dependent density differences between even and odd winding sectors. It develops a semiclassical-to-quantum map by classifying classical open-string solutions and conjecturing the corresponding Hilbert spaces, subsequently verifying aspects via partition-function calculations and NCOS-limit analysis. The NCOS limit reveals noncommutative open-string dynamics on AdS_2, and the work outlines extensions to multi-brane configurations and other branes in AdS_3.

Abstract

We study the spectrum of open strings on AdS_2 branes in AdS_3 in an NS-NS background, using the SL(2,R) WZW model. When the brane carries no fundamental string charge, the open string spectrum is the holomorphic square root of the spectrum of closed strings in AdS_3. It contains short and long strings, and is invariant under spectral flow. When the brane carries fundamental string charge, the open string spectrum again contains short and long strings in all winding sectors. However, branes with fundamental string charge break half the spectral flow symmetry. This has different implications for short and long strings. As the fundamental string charge increases, the brane approaches the boundary of AdS_3. In this limit, the induced electric field on the worldvolume reaches its critical value, producing noncommutative open string theory on AdS_2.

Open Strings on AdS_2 Branes

TL;DR

This paper analyzes open strings ending on AdS_2 branes within AdS_3 in an NS-NS background using the SL(2,R) WZW model. It shows that straight AdS_2 branes yield an open-string spectrum that is the holomorphic square root of the closed-string AdS_3 spectrum, with both short and long strings and full spectral flow, while curved AdS_2 branes break half the spectral flow and introduce ψ_0-dependent density differences between even and odd winding sectors. It develops a semiclassical-to-quantum map by classifying classical open-string solutions and conjecturing the corresponding Hilbert spaces, subsequently verifying aspects via partition-function calculations and NCOS-limit analysis. The NCOS limit reveals noncommutative open-string dynamics on AdS_2, and the work outlines extensions to multi-brane configurations and other branes in AdS_3.

Abstract

We study the spectrum of open strings on AdS_2 branes in AdS_3 in an NS-NS background, using the SL(2,R) WZW model. When the brane carries no fundamental string charge, the open string spectrum is the holomorphic square root of the spectrum of closed strings in AdS_3. It contains short and long strings, and is invariant under spectral flow. When the brane carries fundamental string charge, the open string spectrum again contains short and long strings in all winding sectors. However, branes with fundamental string charge break half the spectral flow symmetry. This has different implications for short and long strings. As the fundamental string charge increases, the brane approaches the boundary of AdS_3. In this limit, the induced electric field on the worldvolume reaches its critical value, producing noncommutative open string theory on AdS_2.

Paper Structure

This paper contains 18 sections, 189 equations, 8 figures.

Table of Contents

  1. Introduction
  2. $S^2$ Branes in the $SU(2)$ WZW Model
  3. Closed Strings in $AdS_3$
  4. Classical Solutions
  5. The Quantum Hilbert Space
  6. The Straight $AdS_2$ Brane
  7. Classical Solutions
  8. The Quantum Hilbert Space
  9. The Curved $AdS_2$ Brane
  10. Classical Solutions
  11. The Quantum Hilbert Space
  12. A Two-Brane System
  13. The NCOS Limit
  14. Summary and Discussion
  15. Coordinate Systems for $AdS_3$
  16. ...and 3 more sections

Figures (8)

  • Figure 1: (a) A view at fixed $t$ of $S^2$ branes and open strings ending on them. (b) A view at fixed $t$ of a system of two $S^2$ branes with open strings.
  • Figure 2: $AdS_2$ branes in $AdS_3$. The view is of the $(\rho, \theta)$ plane at fixed global time $t$. The branes are surfaces of constant $\psi$.
  • Figure 3: (a) A timelike geodesic and (b) a spacelike geodesic confined to the brane at $\psi_0=0$.
  • Figure 4: (a) An open short string obtained from a timelike geodesic by spectral flow with $w=1$. (b) An open long string obtained from a spacelike geodesic by spectral flow with $w=1$.
  • Figure 5: The basic (a) "timelike" and (b) "spacelike" string solutions ending on a curved $AdS_2$ brane.
  • ...and 3 more figures