Krylov complexity for $η$ deformed superstring backgrounds

We compute the holographic Krylov complexity for a class of strongly coupled Quantum Field Theory (QFT) (with broken scale invariance) in a top down approach, where the dual gravitational counterpart corresponds to $η$ deformed superstring solutions in a type IIB set up. The full 10d solution contains the anti-de Sitter ($AdS$) as the subspace, along with the dilaton and the background Ramond-Ramond (RR) and Neveu-Schwarz (NS) fluxes. The Krylov complexity in the dual operator picture is obtained by computing the proper momentum of the massive particle along its geodesic in the bulk spacetime. We explore the effects of $η$ deformation on the bulk momentum of the particle, which in turn affects the rate of growth of complexity in the dual QFTs. Our results boil down into the undeformed case in the appropriate limit of the Yang-Baxter (YB) parameter. We also comment on the complexity of the dual quantum mechanical model with broken scale invariance.