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We prove that the quantum toroidal algebras $\mathcal{E}_\mathbf{s}$associated with different root systems $\mathbf{s}$ of $\mathfrak{gl}_{m|n}$type are isomorphic. We also show the existence of Miki automorphism of$\mathcal{E}_\mathbf{s}$, which exchanges the vertical and horizontalsubalgebras.  To obtain these results, we establish an action of the toroidal braid groupon the direct sum $\oplus_\mathbf{s} \mathcal{E}_\mathbf{s}$ of all suchalgebras.

Braid actions on quantum toroidal superalgebras