Supersymmetry and integrability of the elliptic ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$ superstring

We construct a 1-parameter family of Ramond-Ramond fluxes supporting the elliptic ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$ metric with constant dilaton and preserving 8 of the 16 supercharges of the undeformed background. On the supersymmetric locus, we compute the tree-level worldsheet S-matrix in uniform light-cone gauge up to quadratic order in fermions and find that it non-trivially satisfies the classical Yang-Baxter equation. Moreover, imposing classical integrability and symmetries, we conjecture compatible processes quartic in fermions. We also investigate different limits of interest, including trigonometric deformations and the limit to the ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times {\mathrm{S}}^2\times {\mathrm{T}}^6$ superstring. Our results provide strong evidence for a supersymmetric and integrable elliptic deformation of the ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$ superstring supported by Ramond-Ramond flux and a constant dilaton.