Holographic Krylov Complexity for Charged, Composite and Extended Probes

TL;DR

This paper investigates holographic Krylov complexity for various operators, revealing how internal structure and extension influence complexity growth and subleading effects, thus broadening understanding of operator complexity in holography.

Contribution

It introduces a detailed analysis of complexity for charged, composite, and extended probes, highlighting the effects of internal structure and spatial extension on complexity growth.

Findings

Internal charge modifies complexity growth, enabling symmetry-resolved analysis.

Structured pointlike probes show subleading effects due to internal charges.

Extended operators exhibit different subleading behavior, indicating finer complexity distinctions.

Abstract

We study the holographic spread/Krylov complexity of operators with non-trivial internal structure and of genuinely extended operators. We first consider a massive particle in AdS$_5\times S^5$ carrying conserved $R$-charge, and show how motion in the internal space modifies the complexity growth, yielding a natural holographic realisation of symmetry-resolved Krylov complexity. We then move to probes that are effectively pointlike from the field-theory viewpoint but possess an intrinsic structure in the bulk: baryon-vertex configurations and giant gravitons. Our results indicate that, for this broad class of structured but pointlike probes, the leading large-time behaviour retains the characteristic form expected for local operators in conformal theories, while the internal structure and induced charges produce informative subleading effects. We also study a genuinely extended probe, a fundamental string falling in AdS while stretched along a spatial direction, as a model for the spread complexity of a non-local operator. In this case, although the leading behaviour still exhibits the expected growth pattern, the subleading terms and intermediate regimes differ qualitatively from those of pointlike probes. This provides concrete evidence that extended operators carry a finer notion of spread complexity, sensitive to their spatial structure. Our results broaden the class of probes for which holographic Krylov complexity can be analysed explicitly, clarify which features are universal and which depend on the nature of the operator, and open a promising route toward a sharper field-theory understanding of complexity for charged, composite and extended excitations.