Complexity and Emergence of Warped $\text{AdS}_3$ Space-time from Chiral Liouville Action

Abstract

In this work we explore the complexity path integral optimization process for the case of warped $\text{AdS}_3$/warped $\text{CFT}_2$ correspondence. We first present the specific renormalization flow equations and analyze the differences with the case of CFT. We discuss how the "chiral Liouville action" could replace the Liouville action as the suitable cost function for this case. Starting from the other side of the story, we also show how the deformed Liouville actions could be derived from the spacelike, timelike and null warped metrics and how the behaviors of boundary topological terms creating these metrics, versus the deformation parameter are consistent with our expectations. As the main results of this work, we develop many holographic tools for the case of warped $\text{AdS}_3$, which include the tensor network structure for the chiral warped CFTs, entangler function, surface/state correspondence, quantum circuits of Kac-Moody algebra and kinematic space of WAdS/WCFTs. In addition, we discuss how and why the path-integral complexity should be generalized and propose several other examples such as Polyakov, p-adic strings and Zabrodin actions as the more suitable cost functions to calculate the circuit complexity.