The low-energy limit of AdS$_3$/CFT$_2$ and its TBA

TL;DR

The authors show that in the low-energy, decompactified limit of strings on AdS$_3\times$S$^3\times$T$^4$, the gapless spectrum is captured by massless, relativistic S matrices $S_{LL}$ and $S_{RR}$ for same-chirality scattering. They construct a relativistic Thermodynamic Bethe Ansatz (TBA) to incorporate wrapping corrections and compute the central charge, obtaining $c=6$, consistent with a free 2D CFT (likely four bosons with their superpartners). The massless dressing phase reduces to Zamolodchikov’s sine-Gordon factor in the BMN limit, and the phase aligns with the $\mathcal{N}=2$ super-sine-Gordon structure at a special coupling, reinforcing a two-copy free-theory picture in the IR. Through a detailed ABA construction and a TBA/Y-system analysis, they relate the low-energy AdS$_3$ sector to well-studied integrable models, providing a nonperturbative framework to address finite-size effects and guiding future extensions to the full AdS$_3$ spectral problem and alternative flux backgrounds.

Abstract

We investigate low-energy string excitations in AdS$_3\times$S$^3\times$T$^4$. When the worldsheet is decompactified, the theory has gapless modes whose spectrum at low energies is determined by massless relativistic integrable S matrices of the type introduced by Al. B. Zamolodchikov. The S matrices are non-trivial only for excitations with identical worldsheet chirality, indicating that the low-energy theory is a CFT$_2$. We construct a Thermodynamic Bethe Ansatz (TBA) for these excitations and show how the massless modes' wrapping effects may be incorporated into the AdS$_3$ spectral problem. Using the TBA and its associated Y-system, we determine the central charge of the low energy CFT$_2$ to be $c=6$ from calculating the vacuum energy for antiperiodic fermions - with the vacuum energy being zero for periodic fermions in agreement with a supersymmetric theory - and find the energies of some excited states.

Figures (5)

• Figure 1: The transfer matrix is obtained by identifying $a=b$ in the monodromy matrix, and summing over $\sum_a$. The indices $a$ and $b$ are in the auxiliary $0$-th space, while the indices $c_i$ and $d_i$ pertain to the chain of frame particles (often referred to as the quantum space).
• Figure 2: The Dynkin diagram associated to the Bethe equations (\ref{['betherel1']})-(\ref{['betherel3']}): the central node corresponds to the momentum carrying variable $\theta$, while the crossed nodes denote the fermionic nodes associated to the auxiliary variables $\beta_1$ and $\beta_3$.
• Figure 3: The diagram associated to the TBA equations (\ref{['tba']}): the central node correspond to $\varepsilon_0$, the crossed nodes correspond to the fermionic pseudoenergies $\varepsilon_{\pm n}$ with $n=1,3$ and the lines represent the equations linking the pseudoenergies via the kernel $\phi$.
• Figure 4: Finite solutions chosen for $Y_0(-\infty)$ and $Y_1(-\infty)$ from equations (\ref{['minima']}), and the solution chosen for $Y_0(\infty)$ from (\ref{['maxima']}), in the interval $\gamma\in(0,4\pi)$.
• Figure 5: Comparison of our numerical results with analytic formulas (\ref{['E0pi/2']}), (\ref{['E0pi']}) and (\ref{['par']}), for $R=1$.