Divergences in Holographic Complexity

TL;DR

This paper investigates the UV divergences in the holographic complexity related to the Wheeler-de Witt patch in AdS spacetimes, revealing how boundary terms affect divergence structure and comparing action and volume divergences.

Contribution

It introduces a boundary surface term that removes excessive divergences and compares the divergence structures of action and volume in holographic complexity.

Findings

Including a boundary term makes the leading divergence proportional to boundary volume.

The divergence structure of action and volume are qualitatively similar but differ in subleading terms.

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.