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AdS$_3$ Freelance Holography, A Detailed Analysis

M. M. Sheikh-Jabbari, V. Taghiloo

TL;DR

The paper develops a comprehensive framework for AdS$_3$ freelance holography, enabling the boundary theory to live on arbitrary timelike surfaces with arbitrarily chosen bulk boundary conditions. It provides an explicit radial (FG-like) solution of AdS$_3$ gravity showing the bulk data reduces to two covariantly constant boundary data, and uses the covariant phase space formalism to implement and flow various boundary conditions via a W-term, linking bulk data to boundary multitrace deformations such as TTbar. By systematically classifying covariant and non-covariant boundary conditions, the authors construct diverse solution spaces (Dirichlet/Bañados, Neumann, conformal, conformal-conjugate, generic w, black flowers) and compute their surface charges, revealing Virasoro and u(1) Kac–Moody algebras as appropriate. They also study finite-distance and inside-AdS$_3$ scenarios, including a detailed treatment of Dirichlet, conformal, and Neumann data, with exact radial evolution and RG-like flow of the boundary action, highlighting the role of TTbar in finite-cutoff holography and its holographic interpretation. The work lays a foundation for extending freelance holography to higher dimensions, horizon regions, and matter couplings, and points toward a deeper understanding of boundary-induced gravity and fluid/gravity correspondences within this flexible holographic paradigm.

Abstract

Freelance holography program is an extension of the gauge/gravity correspondence in which the boundary theory can reside on any timelike codimension-one surface in AdS space, and the boundary conditions on the bulk fields can be chosen arbitrarily. Freelance holography provides the framework for a systematic study of various boundary conditions and associated bulk geometries. In this work, we analyze in detail the AdS$_3$ freelance holography. One can explicitly solve for the bulk AdS$_3$ Einstein gravity equations of motion. For generic boundary conditions, the solutions are described by two arbitrary functions of one variable. We study holographic renormalization group (RG) flows, the interpolation between different boundary conditions at different boundaries and associated surface charges and their algebras.

AdS$_3$ Freelance Holography, A Detailed Analysis

TL;DR

The paper develops a comprehensive framework for AdS freelance holography, enabling the boundary theory to live on arbitrary timelike surfaces with arbitrarily chosen bulk boundary conditions. It provides an explicit radial (FG-like) solution of AdS gravity showing the bulk data reduces to two covariantly constant boundary data, and uses the covariant phase space formalism to implement and flow various boundary conditions via a W-term, linking bulk data to boundary multitrace deformations such as TTbar. By systematically classifying covariant and non-covariant boundary conditions, the authors construct diverse solution spaces (Dirichlet/Bañados, Neumann, conformal, conformal-conjugate, generic w, black flowers) and compute their surface charges, revealing Virasoro and u(1) Kac–Moody algebras as appropriate. They also study finite-distance and inside-AdS scenarios, including a detailed treatment of Dirichlet, conformal, and Neumann data, with exact radial evolution and RG-like flow of the boundary action, highlighting the role of TTbar in finite-cutoff holography and its holographic interpretation. The work lays a foundation for extending freelance holography to higher dimensions, horizon regions, and matter couplings, and points toward a deeper understanding of boundary-induced gravity and fluid/gravity correspondences within this flexible holographic paradigm.

Abstract

Freelance holography program is an extension of the gauge/gravity correspondence in which the boundary theory can reside on any timelike codimension-one surface in AdS space, and the boundary conditions on the bulk fields can be chosen arbitrarily. Freelance holography provides the framework for a systematic study of various boundary conditions and associated bulk geometries. In this work, we analyze in detail the AdS freelance holography. One can explicitly solve for the bulk AdS Einstein gravity equations of motion. For generic boundary conditions, the solutions are described by two arbitrary functions of one variable. We study holographic renormalization group (RG) flows, the interpolation between different boundary conditions at different boundaries and associated surface charges and their algebras.

Paper Structure

This paper contains 91 sections, 242 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminaries
  4. AdS3 gravity
  5. Radial decomposition.
  6. AdS$_3$ Einstein gravity action.
  7. Integrating field equations.
  8. 3d freelance holography
  9. Brief review on covariant phase space formalism
  10. Boundary conditions, initial conditions and corner terms
  11. Freelancing: arbitrary boundary conditions at arbitrary constant r surfaces
  12. Symplectic potential flow
  13. Integrating effective action of boundary theory at arbitrary constant r surfaces
  14. A systematic analysis of consistent boundary conditions
  15. Covariant boundary conditions
  16. ...and 76 more sections

Figures (1)

  • Figure 1: Asymptotically AdS$_3$ spacetime and a region cutoff at radius $r$. $\textcolor{black}{\cal{M}}$ denotes the global asymptotically AdS$_3$ spacetime and $\textcolor{blue!80}{\Sigma}$ is its asymptotic timelike boundary. The shaded region $\textcolor{blue!50!black}{\mathcal{M}_r}$ is the part of AdS$_3$ cutoff at radius $r$, enclosed in a timelike surface $\textcolor{darkred!60}{\Sigma_r}$.