Integrable spin chain for stringy Wess-Zumino-Witten models

TL;DR

The paper demonstrates that strings on AdS$_3\times$S$^3\times$T$^4$ with pure NS-NS flux admit a simple integrable spin-chain description in which wrapping corrections cancel and the spectrum is given in closed form. It derives the mirror TBA and shows it reduces to the Bethe-Yang equations, enabling an exact Bethe ansatz solution that reproduces the WZW mass-shell conditions. A detailed mapping between spin-chain magnons and Kač-Moody modes, including spectrally flowed sectors and light-cone winding, establishes a precise equivalence between the spin-chain spectrum and the WZW spectrum. The results bridge AdS$_3$/CFT$_2$ integrability with worldsheet CFT techniques, offering a clean arena to study protected and non-protected correlators and potential extensions to other NS-NS backgrounds and deformations.

Abstract

Building on arXiv:1804.01998 we investigate the integrable structure of the Wess-Zumino-Witten (WZW) model describing closed strings on $AdS_3\times S^3\times T^4$. Using the recently-proposed integrable S matrix we show analytically that all wrapping corrections cancel and that the theory has a natural spin-chain interpretation. We construct the integrable spin chain and discuss its relation with the WZW description. Finally we compute the spin-chain spectrum in closed form and show that it matches the WZW prediction on the nose.

Figures (3)

• Figure 1: Pictorial representation of a wrapping effect. The worldsheet is a cylinder of size $R$. One mirror particle of mirror momentum $\bar{p}(u)$ (dark red) wraps around it, scattering in sequence with particles $1,2\dots K$ (blue). We should sum over all possible mirror particles and integrate over their rapidities $u$.
• Figure 2: Some excitations in the spectrally unflowed sector. We plot in red the dispersion $H(p)$ for the excitation $Y(p)$, and in cyan the one for $\bar{Y}(p)$. The position of the cusp where $kp=-\mu R_{\text{eff}}$ is close to zero---more precisely, it lies in the interval $(-1,+1)$. We highlight the allowed mode numbers with dots; the mode numbers falling on the positive slopes (solid lines) correspond to chiral excitations $J^\pm_{-\nu}$ while the ones on the negative slopes (dashed lines) correspond to anti-chiral ones, $\tilde{J}^\pm_{-\tilde{\nu}}$. The only allowed zero-modes are those of ${J}^-$ and $\tilde{J}^-$, as required by the WZW construction, see also appendix \ref{['app:WZW']}.
• Figure 3: Some excitations in the flowed sector with $w=2$. We plot in red the dispersion $H(p)$ for the excitation $Y(p)$, and in cyan the one for $\bar{Y}(p)$. The position of the cusp where $kp=-\mu R_{\text{eff}}$ is between $-3$ and $-2$, or between $2$ and $3$. We highlight the allowed mode numbers with dots; the mode numbers falling on the positive slopes (solid lines) correspond to chiral excitations $J^\pm_{-\nu}$ while the ones on the negative slopes (dashed lines) correspond to anti-chiral ones, $\tilde{J}^\pm_{-\tilde{\nu}}$. Notice that the mode-number is shifted with respect to figure \ref{['fig:flowedspectrum']}.