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We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidalalgebra $\mathcal E_{m|n}$ associated to Lie superalgebra$\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutativesubalgebras $\mathcal B_{m|n}(s)$ depending on parameters$s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularityconditions. We show that $\mathcal B_{m|n}(s)$ is a free polynomial algebra andgive explicit generators which conjecturally correspond to the traces of the$s$-weighted $R$-matrix computed on the degree zero part of $\mathcal E_{m|n}$modules of levels $\pm 1$.

Commutative subalgebra of a shuffle algebra associated with quantum toroidal $\mathfrak{gl}_{m|n}$