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

Long Strings and Symmetric Product Orbifold from the AdS$_3$ Bethe Equations