Holographic Complexity and Two Identities of Action Growth

TL;DR

This paper demonstrates that the rate of action growth in holographic complexity is equal to the difference in generalized enthalpy between horizons, relying on a key identity relating surface terms to temperature and entropy.

Contribution

It establishes a new connection between action growth and thermodynamic quantities in holography, providing a proof based on horizon surface terms.

Findings

Action growth rate equals the difference of generalized enthalpy between horizons

Surface-term contributions are proportional to temperature and entropy

Provides a theoretical foundation for the complexity-action conjecture

Abstract

The recently proposed complexity-action conjecture allows one to calculate how fast one can produce a quantum state from a reference state in terms of the on-shell action of the dual AdS black hole at the Wheeler-DeWitt patch. We show that the action growth rate is given by the difference of the generalized enthalpy between the two corresponding horizons. The proof relies on the second identity that the surface-term contribution on a horizon is given by the product of the associated temperature and entropy.