Holographic Krylov complexity for Yang-Baxter deformed supergravity backgrounds

TL;DR

This work advances the holographic understanding of Krylov complexity by analyzing top-down Yang-Baxter deformations of type IIB supergravity backgrounds. By mapping operator growth to the bulk radial momentum of a probe particle, it shows that YB deformations generate a finite holographic screen, lead to a nonzero initial momentum, and suppress the growth rate of complexity relative to undeformed AdS; these effects persist across AdS$_2$ and AdS$_3$-factor geometries and in the mirror model. The analysis encompasses one- and two-parameter deformations (κ, κ_L, κ_R) and reveals that special points (e.g., κ_L=κ_R) recover SL(2,$\mathbb{R}$) Krylov sectors, while generic deformations reduce the effective Krylov dimension of the dual QFT. The results provide a concrete, calculable link between bulk geometric deformations and operator growth in strongly coupled quantum field theories, with implications for the interpretation of Krylov complexity in deformed holographic systems.

Abstract

We compute the holographic Krylov complexity for a class of strongly coupled QFTs in a top down approach, where the dual gravitational counterpart corresponds to Yang-Baxter (YB) deformed supergravity solutions in a type IIB set up. The full 10d solution contains the $AdS_n (n=2,3)$ as the subspace, along with the dilaton and the background RR and 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 YB 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 YB parameter.

Figures (3)

• Figure 1: In the Figure above we picturize the geodesic motion of a massive particle (of mass $m$) in the Yang-Baxter (YB) deformed background in global coordinates. The "holographic screen" (which could be thought of as an effective boundary of the spacetime) is located at finite radial distance $\rho_B$, which acts like a radial cut-off in the bulk AdS. Here, $\rho_\infty$ is the location of the true asymptotic boundary of the (undeformed) AdS. From the perspective of an observer sitting at $\rho_\infty$, the massive particle is created at $\rho_\Lambda \sim \rho_B$, which is therefore always at a finite radial distance from the true boundary of the AdS spacetime and is identified with a finite momentum to begin with. The YB deformation reduces the proper geodesic distance of the massive particle trajectory in the bulk and hence the rate of complexity growth for the dual QFT living on $\rho_\infty$.
• Figure 2: We show the rate of growth of Krylov complexity for various choices of the YB deformation parameter ($\kappa$).
• Figure 3: We plot proper momentum against time for different choices of the deformation parameter $\kappa_-=\frac{1}{2}(\kappa_L-\kappa_R)$, where $\kappa_L$ and $\kappa_R$ are close to each other. When they are exactly equal, we have the massive particle experiences an emerging$SL(2,R)$ symmetry which is therefore identical to that of the original notion of the Krylov complexity in a Krylov basis. In our analysis, we set $B=0.1$ (which accounts for the initial velocity ($\dot{\rho}(0)$) of the particle, which is very small to begin with) and $\rho_\Lambda =10$. For $\kappa_- \neq 0$, the initial momenta are very small ($\sim \mathcal{O}(10^{-10})$) and cannot be distinguished from the momenta with $\kappa_-=0$, which is actually zero namely $P_\rho(t=0)|_{\kappa_-=0}=0$.