Complexity and Operator Growth in Holographic 6d SCFTs

Abstract

We study Krylov (spread) complexity in strongly coupled six-dimensional ${\cal N}=(1,0)$ superconformal field theories with holographic duals in massive type IIA supergravity. Extending recent holographic proposals relating Krylov complexity growth to the proper momentum of an infalling particle, we analyse the dynamics of massive geodesic probes in these geometries. In our setup, the bulk particle is allowed to move along three directions: the radial AdS coordinate, the internal $S^2$ associated with the $SU(2)_R$ symmetry, and the coordinate parametrising the quiver. In the dual field theory these motions encode, respectively, operator growth, the presence of R-symmetry charges, and spreading across different nodes of the quiver. We analyse the geodesic motion both analytically and numerically for representative quiver configurations. The motion along the quiver direction is typically damped and localised at early times, while the late-time behaviour is dominated by the radial AdS motion. As a consequence, the generalised proper momentum grows linearly at late times, consistent with expectations for Krylov complexity in conformal theories. The inclusion of angular momentum ($SU(2)_R$ charge) introduces additional constraints on the allowed motion and modifies the early-time dynamics while leaving the asymptotic behaviour unchanged. These results provide a first exploration of Krylov complexity in higher-dimensional holographic conformal theories and reveal how operator growth can probe both internal symmetries and quiver structure in strongly coupled conformal field theories.

Figures (15)

• Figure 1: Linear quiver: balancing implies $F_k=2N_k-N_{k-1}-N_{k+1}$.
• Figure 2: Plot of $f_{1}(\eta)$ for both quiver 1 (left) and quiver 2 (right). We set $P = 100$.
• Figure 3: Trajectory of the particle in $\eta$ for quiver 1. We set $J = 0$, $\eta_{0} = 20$, $P = 100$ and $e^{r_{UV}} = 0.01$.
• Figure 4: Momentum of the particle along the $\eta$ direction (left) and contribution of the $\eta$ momentum to the proper radial momentum. We set $J = 0$, $\eta_{0} = 20$, $P = 100$ and $e^{r_{UV}} = 0.01$.
• Figure 5: Trajectory of the particle in $\eta$ for quiver 1. We set $J = 0$, $\eta_{0} = 93$, $P = 100$ and $e^{r_{UV}} = 0.01$.