Holographic Complexity from the Crofton's Formula in Lorentzian AdS${}_3$

Abstract

We study the Crofton's formula in the Lorentzian AdS${}_3$ and find that the area of a generic space-like two dimensional surface is given by the flux of space-like geodesics. The "complexity=volume" conjecture then implies a new holographic representation of the complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.

23 pages,5 figures,v2: references added

References in corpus (7)

• Quantum Computation as Geometry
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• Surface/State Correspondence as a Generalized Holography
• Einstein's Equations from Varying Complexity
• Path integral optimization as circuit complexity
• Holographic Entanglement of Purification from Conformal Field Theories
• From Path Integrals to Tensor Networks for AdS/CFT