Towards a derivation of holographic entanglement entropy (1102.0440v2)

Abstract: We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

• The paper presents a novel derivation that maps entanglement entropy for spherical surfaces to the thermal entropy of a hyperbolic state via conformal transformation.
• It leverages the AdS/CFT correspondence to relate the boundary CFT state to black hole horizon entropy in the bulk using the Wald entropy formula.
• The study identifies the universal entanglement entropy’s dependence on the A-type anomaly in even dimensions, establishing a measure of quantum degrees of freedom in CFTs.

Overview of "Towards a Derivation of Holographic Entanglement Entropy"

This paper presents a detailed examination of holographic entanglement entropy, focusing on its derivation for spherical entangling surfaces within the context of conformal field theories (CFTs) and gravity duals, particularly using the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence framework. The authors explore a novel approach that deviates from the standard replica trick traditionally used in entanglement entropy calculations.

Key Contributions

1. Conformal Mapping and Entanglement Entropy:
   • The paper introduces a method that connects the entanglement entropy of spherical entangling surfaces in a boundary CFT to the thermodynamic entropy of a thermal state in a hyperbolic geometry $R \times H^{d-1}$. This mapping utilizes a conformal transformation, mapping the vacuum state of the CFT to a thermal state, thus transforming the problem into one calculable via thermodynamics.
2. Holographic Calculations:
   • The research aligns this approach with holography by employing the AdS/CFT correspondence, where the thermal state of the boundary CFT corresponds to a black hole in the bulk AdS space. The horizon entropy of this topological black hole, given by the Wald entropy formula, then translates back to the entanglement entropy of the boundary theory.
3. Central Charge and Entropy:
   • For even-dimensional boundaries, the authors demonstrate that the universal part of the entanglement entropy is precisely determined by the $A$-type trace anomaly of the CFT. This aligns the universal contribution to the entanglement entropy with the central charge, offering a robust measure of the quantum degrees of freedom in the boundary theory.
4. Generality and Extensions:
   • The derivation is extended to CFTs in both flat and cylindrical geometries, indicating the flexibility and potential of the approach across different conformal mappings. This not only strengthens the theoretical understanding but also verifies the consistency of holographic proposals across different geometrical backgrounds and constraints.

Implications and Future Directions

• The alignment of holographic entanglement entropy with thermodynamic entropy via conformal mappings offers a promising alternative framework that can bypass the limitations of the replica trick, particularly in complex or high-curvature contexts.
• The connection between the leading contribution to entanglement entropy and the $A$-type anomaly in even dimensions offers a metric for evaluating the number of degrees of freedom in quantum theories, potentially extending the use of entanglement entropy as a heuristic for probing the structure of quantum field theories.
• The method suggests pathways to explore even more generalized behaviors of entanglement entropy in varied holographic theories, potentially including higher curvature and non-local gravity theories, where traditional methods encounter limitations.
• Further exploration might involve applying this understanding to other shapes of entangling surfaces, such as elliptic or fractal geometries, and probing the relationship between these geometries and the subtleties of quantum entanglement within the AdS/CFT framework.

This research solidifies the conceptual and practical linkages between entanglement entropy, conformal field theory, and their holographic duals, advancing the field’s comprehension of quantum gravity phenomena. It underscores the transformative potential of conformal methods in unlocking the entropic nature of space-time and its geometric entanglements.