Divergences in Holographic Complexity
Alan Reynolds, Simon F. Ross
TL;DR
This paper analyzes UV divergences in holographic complexity under the CV and CA prescriptions. By incorporating the null-boundary counterterm ΔS from recent work, the CA action becomes coordinate-invariant and its leading UV divergence aligns with the CV result, scaling with the boundary volume. The authors extend the analysis to subleading divergences using Fefferman-Graham expansion, showing these terms are geometry-dependent power laws with different coefficients for CV and CA. Global AdS examples illustrate that, despite a shared leading behavior, CV and CA are distinct in cutoff dependence and subleading structure, highlighting the sensitivity to how the Wheeler-DeWitt patch is cut. Overall, the work supports a common leading-order picture while clarifying quantitative differences between the two complexity notions.
Abstract
We study the UV divergences in the action of the "Wheeler-de Witt patch" in asymptotically AdS spacetimes, which has been conjectured to be dual to the computational complexity of the state of the dual field theory on a spatial slice of the boundary. We show that including a surface term in the action on the null boundaries which ensures invariance under coordinate transformations has the additional virtue of removing a stronger than expected divergence, making the leading divergence proportional to the proper volume of the boundary spatial slice. We compare the divergences in the action to divergences in the volume of a maximal spatial slice in the bulk, finding that the qualitative structure is the same, but subleading divergences have different relative coefficients in the two cases.