On quantum deformations of $AdS_3 \times S^3 \times T^4$ and mirror duality

We consider various integrable two-parameter deformations of the $AdS_3 \times S^3 \times T^4$ superstring with quantum group symmetry. Working on the string worldsheet in light-cone gauge and to quadratic order in fermions, we obtain their common massive tree-level two-body S matrix, which matches the expansion of the conjectured exact $q$-deformed S matrix. We then analyze the behavior of the exact S matrix under mirror transformation -- a double Wick rotation on the worldsheet -- and find that it satisfies a mirror duality relation analogous to the distinguished $q$-deformed $AdS_5 \times S^5$ S matrix in the one parameter deformation limit. Finally, we show that the fermionic $q$-deformed $AdS_5 \times S^5$ S matrix also satisfies such a relation.