Wedge Holographic Complexity in Karch-Randall Braneworld

TL;DR

This paper explores holographic complexity in wedge-shaped AdS geometries with branes, analyzing the effects of fluctuations and intrinsic gravity on complexity growth rates using CV and CA proposals.

Contribution

It provides a detailed comparison of complexity growth in wedge holography from bulk, brane, and boundary perspectives, including effects of brane fluctuations and intrinsic gravity.

Findings

Late-time growth rates match across perspectives for tensionless branes.

Fluctuations cause specific corrections to complexity growth, with CV more sensitive to geometric details.

Additional constant term in CA complexity due to edge mode effects.

Abstract

We investigate holographic complexities in the context of wedge holography, focusing specifically on black string geometry in AdS$_3$. The wedge spacetime is bounded by two end-of-the-world (EOW) branes with intrinsic Dvali-Gabadadze-Porrati (DGP) gravity. In line with this codimension-two holography, there are three equivalent perspectives: bulk perspective, brane perspective, and boundary perspective. Using both the ''Complexity=Volume'' (CV) and ''Complexity=Action'' (CA) proposals, we analyze the complexity in wedge black string geometry in the tensionless limit. By treating the branes as rigid, we find the late-time growth rates of CV and CA match exactly across bulk and brane perspectives. These results are consistent with those from JT gravity, with additional contributions from the intrinsic gravity of the branes. For fluctuating branes, we find that the late-time growth rates of CV and CA match between bulk and brane perspectives at the linear order of fluctuation. The CV results exhibit $\frac{\phi_h^2}{\phi_0}$ corrections from fluctuations, consistent with the findings in previous work. Moreover, the CA results reveal an additional constant term in the fluctuating branes case. We provide an interpretation of this in terms of gravitational edge mode effects. The distinct corrections arising from fluctuations in the CA and CV proposals suggest that the CV proposal is more sensitive to geometric details. Furthermore, we discuss these results in relation to Lloyd's bound on complexity, their general time dependence, and the effects of fluctuations.