Comments on Integrability in the Symmetric Orbifold

We present a map between the excitation of the symmetric-product orbifold CFT of $T^4$, and of the worldsheet-integrability description of $AdS_3\times S^3\times T^4$ of Lloyd, Ohlsson Sax, Sfondrini, and Stefa\'nski at $k=1$. We discuss the map in the absence of RR fluxes, when the theory is free, and at small RR flux, $h\ll 1$, where the symmetric-orbifold CFT is deformed by a marginal operator from the twist-two sector. We discuss the recent results of Gaberdiel, Gopakumar, and Nairz, who computed from the perturbed symmetric-product orbifold the central extension to the symmetry algebra of the theory and its coproduct. We show that it coincides with the $h\ll 1$ expansion of the lightcone symmetry algebra known from worldsheet integrability, and that hence the S matrix found by Gaberdiel, Gopakumar, and Nairz maps to the one bootstrapped by the worldsheet integrability approach.