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

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.