Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lecture notes on strings in AdS$_3$ from the worldsheet and the AdS$_3$/CFT$_2$ duality

Nicolas Kovensky

TL;DR

These notes provide a comprehensive worldsheet treatment of strings in $AdS_3$ with NSNS flux, focusing on exact solvability of the SL$(2,\mathbb{R})$ WZW model and the crucial role of spectral flow in generating the full physical spectrum. They develop both unflowed and spectrally flowed sectors, employing Wakimoto free-field realizations and covering-map techniques to compute correlation functions, and they connect these results to holographic expectations in the boundary CFT, including the D1-D5 system and the tensionless limit at $k=3$. A key accomplishment is the explicit construction of spacetime symmetry generators from worldsheet currents, the derivation of exact two- and three-point functions, and the formulation of recursive and $y$-basis methods that organize higher-point flowed correlators. In the tensionless regime, the localization of worldsheet correlators on covering maps yields a precise link to symmetric orbifold holography and highlights a rich interplay between higher-spin massless modes and holographic duals, with extensions to the exact chiral ring in the supersymmetric setting. Overall, the notes provide a detailed, technically rich framework for understanding AdS$_3$/CFT$_2$ holography in NSNS backgrounds, including exact results, conjectures, and precise matches to protected boundary data.

Abstract

These lecture notes, based on the course given at IPhT in November/December 2023, provide a pedagogical introduction to the study of strings in AdS$_3$ backgrounds supported by NSNS flux from the worldsheet perspective, including a number of updates incorporating recent results. We attempt to give a self-contained overview of the state-of-the-art understanding of this topic, describing key aspects of its 25-year-long rich history alongside some important recent developments, with an emphasis on the computation of worldsheet correlation functions involving spectrally flowed insertions.

Lecture notes on strings in AdS$_3$ from the worldsheet and the AdS$_3$/CFT$_2$ duality

TL;DR

These notes provide a comprehensive worldsheet treatment of strings in with NSNS flux, focusing on exact solvability of the SL WZW model and the crucial role of spectral flow in generating the full physical spectrum. They develop both unflowed and spectrally flowed sectors, employing Wakimoto free-field realizations and covering-map techniques to compute correlation functions, and they connect these results to holographic expectations in the boundary CFT, including the D1-D5 system and the tensionless limit at . A key accomplishment is the explicit construction of spacetime symmetry generators from worldsheet currents, the derivation of exact two- and three-point functions, and the formulation of recursive and -basis methods that organize higher-point flowed correlators. In the tensionless regime, the localization of worldsheet correlators on covering maps yields a precise link to symmetric orbifold holography and highlights a rich interplay between higher-spin massless modes and holographic duals, with extensions to the exact chiral ring in the supersymmetric setting. Overall, the notes provide a detailed, technically rich framework for understanding AdS/CFT holography in NSNS backgrounds, including exact results, conjectures, and precise matches to protected boundary data.

Abstract

These lecture notes, based on the course given at IPhT in November/December 2023, provide a pedagogical introduction to the study of strings in AdS backgrounds supported by NSNS flux from the worldsheet perspective, including a number of updates incorporating recent results. We attempt to give a self-contained overview of the state-of-the-art understanding of this topic, describing key aspects of its 25-year-long rich history alongside some important recent developments, with an emphasis on the computation of worldsheet correlation functions involving spectrally flowed insertions.
Paper Structure (60 sections, 716 equations, 6 figures)

This paper contains 60 sections, 716 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Geometry, spectrum and spacetime symmetries
  3. Geometric aspects and semiclassical considerations
  4. Wakimoto fields and spacetime symmetries
  5. Spectrally flowed representations
  6. Wakimoto fields and $x$-basis spectrally flowed vertex operators
  7. Comments on the partition function
  8. Exact description of the unflowed sector
  9. Reflection symmetry. Two- and three-point functions
  10. OPE structure and four-point functions
  11. Spacetime symmetry generators
  12. Correlation functions and spectral flow
  13. $m$-basis flowed correlators and parafermions
  14. Spectral flow selection rules
  15. Spectrally flowed correlators and covering maps
  16. ...and 45 more sections

Figures (6)

  • Figure 1: From left to right: Particle geodesics (a) timelike solution with $\rho_0=0$ (b) more general timelike configuration (c) basic spacelike solution (d) more general spacelike configuration.
  • Figure 2: Short string (left) and long string (right) solutions with spectral flow $\omega=1$.
  • Figure 3: Geodesic for an unflowed two-point function (left) and classical configuration associated with a four-point function involving operators with non-trivial spectral flow charges (right).
  • Figure 4: Weight diagram for the lowest-weight representation $\hat{{\cal{D}}}_j^+$
  • Figure 5: Weight diagram for the lowest-weight representation $\hat{{\cal{D}}}_j^{+,\omega}$ with $\omega=1$.
  • ...and 1 more figures