Comments on Holographic Complexity

TL;DR

This paper analyzes the structure of UV divergences in holographic complexity conjectures, revealing their geometric nature and exploring extensions to subregion complexity with new divergence features.

Contribution

It provides a detailed analysis of UV divergences in holographic complexity and proposes extensions to subregion complexity, highlighting their geometric structure and new divergence types.

Findings

UV divergences have local geometric integral coefficients

Subregion complexity exhibits similar divergences with additional surface-related divergences

Implications of geometric UV divergences are discussed

Abstract

We study two recent conjectures for holographic complexity: the complexity=action conjecture and the complexity=volume conjecture. In particular, we examine the structure of the UV divergences appearing in these quantities, and show that the coefficients can be written as local integrals of geometric quantities in the boundary. We also consider extending these conjectures to evaluate the complexity of the mixed state produced by reducing the pure global state to a specific subregion of the boundary time slice. The UV divergences in this subregion complexity have a similar geometric structure, but there are also new divergences associated with the geometry of the surface enclosing the boundary region of interest. We discuss possible implications arising from the geometric nature of these UV divergences.