Inequivalent light-cone gauge-fixings of strings on $Ad{S}_n\times S^n$ backgrounds

Light-cone gauge-fixed sigma-models on $Ad{S}_n\times S^n$ backgrounds play an important role in the integrability formulation of the AdS/CFT correspondence. The string spectrum of the sigma-model is gauge-independent, however the Hamiltonian 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 $T\bar{T}$, $\tilde{J}{T}_{\tau }$, $J{T}_{\sigma }$ and $J^{\tau }$ deformations. We investigate the moduli space of equivalent light-cone gauges and, specialising to $Ad{S}_5\times S^5$, compute the different light-cone gauge symmetry algebras, well-known to be $\mathfrak{p}\mathfrak{s}\mathfrak{u}(2\mathrm{\mid }2{)}^{\oplus 2}\oplus \mathfrak{u}(1{)}^{\oplus 2}$ for the standard gauge-fixing. Many integrable deformations require a non-standard light-cone gauge, hence our classification and analysis of inequivalent gauges will be important for analysing such models.