[Paper Review] Long Strings and Symmetric Product Orbifold from the AdS$_3$ Bethe Equations

Alessandro Sfondrini|arXiv (Cornell University)|Oct 6, 2020

Black Holes and Theoretical Physics1 references4 citations

TL;DR

This paper demonstrates that the AdS₃ Bethe equations for the pure-NSNS flux background yield a discrete spectrum at level k=1, exactly matching the partition function of the symmetric product orbifold CFT Sym^Nc(T⁴). By identifying a new class of long-string continuum solutions for k≥2 and showing their disappearance at k=1, the work establishes a precise integrability-based derivation of the orbifold CFT spectrum from string worldsheet dynamics.

ABSTRACT

A particularly rich class of integrable systems arises from the AdS/CFT duality. There, the two-dimensional quantum field theory living on the string worldsheet may be understood in terms of a non-relativistic factorized S matrix, and the energy spectrum may be derived by techniques such as the mirror thermodynamic Bethe ansatz or the quantum spectral curve. In the case of AdS$_3$/CFT$_2$ without Ramond-Ramond fluxes, the worldhseet theory is a Wess-Zumino-Witten model with continous and discrete representations which, for the lowest allowed level, is dual to the symmetric product orbifold of a free theory. I will show how continuous representations may arise from integrability, and that at lowest level the Bethe equations yield the symmetric product orbifold partition function on the nose.

Motivation & Objective

• To establish a direct link between the integrable structure of the AdS₃ string worldsheet and the symmetric product orbifold CFT at level k=1.
• To identify and analyze a new class of long-string continuum solutions in the Bethe equations for k≥2.
• To show that at k=1, the Bethe equations yield a discrete spectrum matching the orbifold partition function exactly.
• To clarify the role of chirality and zero modes in the emergence of the orbifold spectrum from integrability.
• To explore the implications of the absence of RR flux for the integrable structure and spectral properties.

Proposed method

• Deriving the worldsheet S-matrix and Bethe equations for the AdS₃×S³×T⁴ string in pure-NSNS flux background.
• Analyzing the Bethe equations to identify exceptional solutions corresponding to long-string continuum states for k≥2.
• Constructing BPS states from oscillator excitations on the Ramond vacuum with specific mode assignments and R-charge constraints.
• Imposing the orbifold-invariance condition ∑nⱼ ≡ ∑ñⱼ mod w to reproduce the level-matching condition in the orbifold CFT.
• Using the auxiliary su(2)ₐ symmetry to construct singlet states and verify quantum numbers (J³ = w/2, H = 0).
• Matching the resulting partition function to the known generating function F_w(q) of the symmetric product orbifold.

Experimental results

Research questions

• RQ1 How do long-string continuum solutions in the AdS₃ Bethe equations relate to the worldsheet integrability structure at k≥2?
• RQ2 What happens to the spectrum of the AdS₃ string theory when the level k is reduced to 1, particularly in the absence of RR flux?
• RQ3 Can the Bethe equations reproduce the exact partition function of the symmetric product orbifold CFT Sym^Nc(T⁴) at k=1?
• RQ4 What is the role of chirality and zero modes in the emergence of the discrete BPS spectrum at k=1?
• RQ5 How do finite-volume effects and wrapping corrections influence the spectrum at k≥2, and can they generate a gap?

Key findings

• The Bethe equations for the AdS₃ string in pure-NSNS flux exhibit a long-string continuum for k≥2, indicating a continuous spectrum.
• At k=1, the long-string solutions disappear and the spectrum becomes discrete, matching the BPS spectrum of the symmetric product orbifold CFT.
• The discrete spectrum at k=1 is constructed from oscillator excitations on the Ramond vacuum with specific mode assignments and R-charge constraints.
• The resulting partition function precisely matches the generating function F_w(q) of the symmetric product orbifold, including the level-matching condition via ∑nⱼ ≡ ∑ñⱼ mod w.
• The BPS state |Σ_w⟩ has J³ = w/2 and H = 0, confirming its identification with the orbifold's single-cycle state.
• The absence of RR flux is essential: even a small h>0 would break chirality and destroy the long-string continuum and the exact match at k=1.