AdS/BCFT: Holography with Boundaries

• AdS/BCFT correspondence is a theoretical framework that extends holography by including boundaries in conformal field theories.
• It models gravitational duals using end-of-the-world branes and mixed boundary conditions, enabling evaluations of boundary entropy and g-functions.
• The approach finds applications in quantum Hall systems, modified gravity, and fluid dynamics, offering a bridge between holographic models and boundary phenomena.

The AdS/BCFT correspondence is an intriguing theoretical framework that extends the well-known AdS/CFT correspondence by considering boundary effects in conformal field theories. This correspondence provides new insights and tools for understanding how holography can represent systems with boundaries, leading to implications in areas such as quantum field theory, condensed matter physics, and quantum gravity.

1. Conceptual Foundation

The core idea of the AdS/BCFT correspondence is to generalize the AdS/CFT framework to scenarios where the conformal field theory (CFT) is defined on a space with boundaries, referred to as boundary conformal field theories (BCFTs). In this setting, the boundaries are extended into the bulk of an asymptotically Anti-de Sitter (AdS) space via additional boundary hypersurfaces often referred to as "end-of-the-world" (EOW) branes. These branes are governed by Neumann boundary conditions, which impose constraints on their geometry, contrasting the typical Dirichlet conditions of AdS/CFT.

2. Holographic Duals

AdS/BCFT allows the construction of gravitational duals for BCFTs where the geometry in the AdS space reflects the presence of boundaries in the dual field theory. Various setups have been proposed, such as configurations involving strips, disks, and time-dependent boundaries. The main endeavor is to embed the boundary effects of the BCFT into a larger (d+1)-dimensional space, enabling holographic calculations of quantities like the boundary entropy or g-function. This involves using mixed boundary conditions on both the original AdS space and the newly introduced brane.

3. Boundary Entropy and the g-function

Central to the paper of BCFTs is the concept of boundary entropy, which represents degrees of freedom associated uniquely with the boundary. In holography, this is captured by the g-function, often defined as the ground state degeneracy factor. Calculating the g-function holographically involves evaluating the entanglement entropy using partition functions or minimal surfaces. The boundary entropy is shown to agree with known field theory results, providing robust checks of the AdS/BCFT correspondence.

4. Extension to Quantum Hall Systems

AdS/BCFT has been adapted to describe systems like quantum Hall states. These systems are characterized by quantized Hall conductance and edge modes, which can be modeled holographically using configurations of charged black holes and gauge fields. One central finding is the constrained relation between quantum Hall conductance and boundary conditions set by Neumann conditions on the gauge fields. This highlights how topological features can be encoded in holographic models.

5. Thermodynamics in Extended Gravity

The correspondence has been extended to modified gravity theories such as Horndeski and Lovelock. These setups explore the impact of scalar fields and higher-order corrections on the thermodynamics of black holes in the holographic model. Boundary entropy calculations, phase transitions like the Hawking-Page transition, and conformal symmetry restoration at high temperatures exemplify these investigations. Such models provide insights into how altered gravitational dynamics might affect the dual field theory.

6. Connection with Fluid Dynamics

Utilizing the fluid/gravity correspondence, AdS/BCFT contributes to understanding conformal fluid boundary conditions. By translating metric boundary conditions on the EOW brane into constraints on BCFT fluid dynamics, insights into velocity and temperature fields are gleaned. This exploration underscores how gravitational dynamics in the bulk translate into hydrodynamic behavior in the boundary theory, highlighting constraints like no-slip and no-penetration conditions within fluid models.

7. Surface Growth and Bulk Reconstruction

Recent developments propose using surface growth approaches for reconstructing bulk geometry in AdS/BCFT. This involves geometrical growth of extremal surfaces that naturally encode the boundary conditions of the BCFT. The connection of these surfaces with tensor network constructs like MERA offers a deeper understanding of the entanglement structures and the emergent nature of spacetime in holographic duals.

In conclusion, AdS/BCFT correspondence enriches the holographic framework by incorporating boundary effects, leading to profound implications across quantum theories. Its versatility in describing boundary phenomena and extending traditional holographic techniques underscores its importance for future theoretical explorations and applications in quantum systems where boundary conditions play a critical role.