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BRST Quantization of String Theory in AdS(3)

Ari Pakman

TL;DR

This work extends BRST quantization to bosonic and NSR strings on $AdS_3 \times N$ by applying the Frenkel-Garland-Zuckerman framework. By decomposing the worldsheet theory into a parafermionic $SL(2,\mathbb{R})/U(1)$ coset plus a timelike sector and including spectrally flowed representations, the authors establish a vanishing theorem that allows a trace-based no-ghost proof without constructing physical states explicitly. They compute signatures and indices across bosonic and fermionic sectors, including NS and Ramond cases, and show that the BRST cohomology at ghost number zero contains only positive-norm states, even with flowed sectors. The results strengthen the consistency of string propagation in curved $AdS_3$ backgrounds and open avenues for applying FGZ techniques to other gauged WZW and coset geometries in the AdS/CFT context.

Abstract

We study the BRST quantization of bosonic and NSR strings propagating in AdS(3) x N backgrounds. The no-ghost theorem is proved using the Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed representations of affine SL(2,R) appear on an equal footing. Possible generalizations to related curved backgrounds are discussed.

BRST Quantization of String Theory in AdS(3)

TL;DR

This work extends BRST quantization to bosonic and NSR strings on by applying the Frenkel-Garland-Zuckerman framework. By decomposing the worldsheet theory into a parafermionic coset plus a timelike sector and including spectrally flowed representations, the authors establish a vanishing theorem that allows a trace-based no-ghost proof without constructing physical states explicitly. They compute signatures and indices across bosonic and fermionic sectors, including NS and Ramond cases, and show that the BRST cohomology at ghost number zero contains only positive-norm states, even with flowed sectors. The results strengthen the consistency of string propagation in curved backgrounds and open avenues for applying FGZ techniques to other gauged WZW and coset geometries in the AdS/CFT context.

Abstract

We study the BRST quantization of bosonic and NSR strings propagating in AdS(3) x N backgrounds. The no-ghost theorem is proved using the Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed representations of affine SL(2,R) appear on an equal footing. Possible generalizations to related curved backgrounds are discussed.

Paper Structure

This paper contains 19 sections, 107 equations.

Table of Contents

  1. Introduction
  2. The affine $\mathop{\rm SL}\nolimits(2,\mathbb{R})$ theory
  3. The bosonic algebra
  4. $\mathop{\rm SL}\nolimits(2,\mathbb{R})/\mathop{\rm {}U}\nolimits(1) \times \mathop{\rm {}U}\nolimits(1)$ decomposition and spectral flow
  5. The supersymmetric algebra
  6. The Hodge theory of the BRST complex
  7. The FGZ proof of the no-ghost theorem
  8. No-ghost theorem for strings in $AdS_3$
  9. Bosonic strings
  10. Fermionic strings
  11. Neveu-Schwarz sector.
  12. Ramond sector.
  13. Discussion
  14. Note added.
  15. BRST essentials
  16. ...and 4 more sections