Abstract
In this work we explore the complexity path integral optimization process for the case of warped AdS3/warped CFT2 correspondence. We first present the specific renor- malization 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 AdS3, 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.
Highlights
To understand better the interplay between information and geometry, mathematical tools such as kinematic space in the setup of integral geometry in holography was developed in [3,4,5]
We start from a spacelike warped AdS metric and using Topologically Massive Gravity (TMG) and the procedure introduced in [30], we derive a form of a deformed chiral Liouville action and we study its properties
In this work we studied the optimization path integral complexity for the case of warped conformal field theory in the setup of WAdS3/WCFT2
Summary
We first review warped conformal field theories (CFTs) and the chiral Liouville gravity. The connections between local and global symmetries in the boundary and bulk in the AdS/CFT setup, their exact definitions and the interplay between them have been recently studied in more details in [33, 34] Their results would help to construct the tensor network model for each of these deformed CFTs. We would like to see how lack of this symmetry would change the optimization procedure proposed in [11, 12] and how it would change the properties of computational complexity. For the chiral case though W (n − n ) = W (n − n), but as these weights satisfy the Yang-Baxter equation, the model would be integrable Considering these real-world systems and their applications, would make studying various quantum information properties of these models, such as holographic complexity or their models of tensor network much more interesting, even from a practical point of view. Note that the above geometries appear in the near-horizon limit of fourdimensional extremal Kerr black holes where at fixed polar angle the three-dimensional warped AdS3 would appear, leading to the Kerr/CFT correspondence
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