What kind of complexity is dual to holographic complexity?

It is assumed that the holographic complexities such as the complexity-action (CA) and the complexity-volume (CV) conjecture are dual to complexity in field theory. However, because the definition of the complexity in field theory is still not complete, the confirmation of the holographic duality of the complexity is ambiguous. To improve this situation, we approach the problem from a different angle. We first identify minimal and genuin properties that the filed theory dual of the holographic complexity should satisfy without assuming anything from the circuit complexity or the information theory. Based on these properties, we propose a field theory formula dual to the holographic complexity. Our field theory formula implies that the complexity between certain states in two dimensional CFTs is given by the Liouville action, which is compatible with the path-integral complexity. It gives natural interpretations for both the CA and CV conjectures and identify what their reference states are. When applied to the thermo-field double states, it also gives consistent results with the holographic results in the CA conjecture: both the divergent term and finite term.

Automated summary

This paper discusses the duality between holographic complexity and field theory complexity, proposing a field theory formula based on specific properties to describe holographic complexity, which is consistent with path-integral complexity and provides a clear interpretation of the reference states in complexity conjectures.