Complexity and Operator Growth in Holographic 6d SCFTs

TL;DR

This paper investigates Krylov complexity in 6d superconformal field theories with holographic duals, analyzing geodesic probes to understand operator growth, symmetry effects, and quiver spreading.

Contribution

It extends holographic complexity analysis to higher-dimensional theories, exploring how internal symmetries and quiver structures influence operator growth.

Findings

Late-time growth of proper momentum is linear, indicating operator growth.

Angular momentum modifies early-time dynamics but not asymptotic behavior.

Operator spreading encodes internal symmetries and quiver structure in strongly coupled theories.

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.