Quantization of the $N{=}2$ supersymmetric $\text{KdV}$ hierarchy

We continue the study of the quantization of supersymmetric integrable KdV hierarchies. We consider the $N{=}2$ KdV model based on the $sl^{(1)}(2\,|\,1)$ affine algebra but with a new algebraic construction for the $L$-operator, different from the standard Drinfeld–Sokolov reduction. We construct the quantum monodromy matrix satisfying a special version of the reflection equation and show that in the classical limit, this object precisely gives the monodromy matrix of the $N{=}2$ supersymmetric KdV system. We also show that at both the classical and the quantum levels, the trace of the monodromy matrix {(}transfer matrix{\rm)} is invariant under two supersymmetry transformations and the zero mode of the associated $U(1)$ current.