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Light-cone gauge-fixed sigma-models on $AdS_n\times S^n$ backgrounds play animportant role in the integrability formulation of the AdS/CFT correspondence.The string spectrum of the sigma-model is gauge-independent, however theHamiltonian and scattering matrix of the transverse worldsheet fields are not.We study how these change for a large family of inequivalent light-cone gauges,which are interpreted as $T\bar{T}$, $\tilde{J}T_\tau$, $JT_\sigma$ and$J^\tau$ deformations. We investigate the moduli space of equivalent light-conegauges and, specialising to $AdS_5 \times S^5$, compute the differentlight-cone gauge symmetry algebras, well-known to be$\mathfrak{psu}(2|2)^{\oplus 2} \oplus \mathfrak{u}(1)^{\oplus 2}$ for thestandard gauge-fixing. Many integrable deformations require a non-standardlight-cone gauge, hence our classification and analysis of inequivalent gaugeswill be important for analysing such models.

Inequivalent light-cone gauge-fixings of strings on $AdS_n \times S^n$ backgrounds