Holographic Krylov complexity in $\mathcal{N}=4$ SYM

We propose and calculate a holographic Krylov complexity in $\mathcal{N}=4$ SYM via the proper momentum for motion in $Ad{S}_5$ sliced by $Ad{S}_3$. The motion in an $Ad{S}_3$ subgroup corresponds to the Krylov complexity of the $Sl(2)$ subsector. The general motion corresponds to the Krylov complexity of the $\mathcal{N}=4$ SYM.