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Strings in AdS_3 and the SL(2,R) WZW Model. Part 1: The Spectrum

Juan Maldacena, Hirosi Ooguri

TL;DR

This paper proposes that the SL(2,R) WZW model for AdS_3 requires including spectral-flowed representations to obtain a unitary, ghost-free string spectrum. By combining continuous and discrete SL(2,R) representations with all integer spectral flows, Maldacena and Ooguri classify short and long string sectors and show how long strings correspond to flowed continuous reps. They establish a no-ghost theorem within the flowed framework and derive the physical spectrum, including discrete spacetime energies for flowed discrete states and boundary-like long-string states, connecting to scattering and to the SL(2,R)/U(1) coset. The work lays a foundation for exact AdS_3 string theory and its holographic implications, providing tools for modular invariance, vertex operators, and long-string S-matrix computations.

Abstract

In this paper we study the spectrum of bosonic string theory on AdS_3. We study classical solutions of the SL(2,R) WZW model, including solutions for long strings with non-zero winding number. We show that the model has a symmetry relating string configurations with different winding numbers. We then study the Hilbert space of the WZW model, including all states related by the above symmetry. This leads to a precise description of long strings. We prove a no-ghost theorem for all the representations that are involved and discuss the scattering of the long string.

Strings in AdS_3 and the SL(2,R) WZW Model. Part 1: The Spectrum

TL;DR

This paper proposes that the SL(2,R) WZW model for AdS_3 requires including spectral-flowed representations to obtain a unitary, ghost-free string spectrum. By combining continuous and discrete SL(2,R) representations with all integer spectral flows, Maldacena and Ooguri classify short and long string sectors and show how long strings correspond to flowed continuous reps. They establish a no-ghost theorem within the flowed framework and derive the physical spectrum, including discrete spacetime energies for flowed discrete states and boundary-like long-string states, connecting to scattering and to the SL(2,R)/U(1) coset. The work lays a foundation for exact AdS_3 string theory and its holographic implications, providing tools for modular invariance, vertex operators, and long-string S-matrix computations.

Abstract

In this paper we study the spectrum of bosonic string theory on AdS_3. We study classical solutions of the SL(2,R) WZW model, including solutions for long strings with non-zero winding number. We show that the model has a symmetry relating string configurations with different winding numbers. We then study the Hilbert space of the WZW model, including all states related by the above symmetry. This leads to a precise description of long strings. We prove a no-ghost theorem for all the representations that are involved and discuss the scattering of the long string.

Paper Structure

This paper contains 25 sections, 134 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Classical solutions
  3. Geodesics in AdS$_{\bf 3}$
  4. Spectral flow and strings with winding numbers
  5. Short strings as spectral flow of timelike geodesics
  6. Long strings as spectral flow of spacelike geodesics
  7. Semi-classical analysis
  8. Quantum string in AdS$_{\bf 3}$
  9. Representations of the zero modes
  10. Representations of the current algebra and no-ghost theorem
  11. Spectral flow and the long string
  12. Physical spectrum
  13. Scattering of long string
  14. Vertex operators
  15. Reflection coefficient
  16. ...and 10 more sections

Figures (7)

  • Figure 1: Timelike geodesic; (A) a solution (\ref{['timelikegeodesic']}) with $U=V=1$, (B) a general geodesic is obtained by acting the $SL(2,R) \times SL(2,R)$ isometry on (A).
  • Figure 2: Spacelike geodesic; (A) a solution (\ref{['spacelikegeodesic']}) with $U=V=1$, (B) a general geodesic is obtained by acting the $SL(2,R) \times SL(2,R)$ isometry on (A).
  • Figure 3: A classical solution obtained by the spectral flow of a timelike geodesic. The solution repeats expansion and contraction. The maximum size of the string is $\rho = \rho_0$.
  • Figure 4: A long string solution obtained by the spectral flow of a spacelike geodesic. The long string comes from the boundary of $AdS_3$, collapse to a point and then expands away to the boundary of $AdS_3$ again.
  • Figure 5: Weight diagram the representation $\widehat{\cal D}^{+}_j$, whose the primary states form a discrete lowest weight representation ${\cal D}_j^+$.
  • ...and 2 more figures