Worldline approach to semi-classical conformal blocks (1501.02260v3)

Abstract: We extend recent results on semi-classical conformal blocks in 2d CFT and their relation to 3D gravity via the AdS/CFT correspondence. We consider four-point functions with two heavy and two light external operators, along with the exchange of a light operator. By explicit computation, we establish precise agreement between these CFT objects and a simple picture of particle worldlines joined by cubic vertices propagating in asymptotically AdS$_3$ geometries (conical defects or BTZ black holes). We provide a simple argument that explains this agreement.

• The paper introduces a worldline-based approach combined with monodromy methods to compute semi-classical conformal blocks in 2D CFT.
• The technique reproduces 3D AdS gravity dual predictions by analyzing particle geodesics in conical defects and BTZ black hole geometries.
• This approach paves the way for future research on higher-point functions and quantum corrections beyond the semi-classical regime.

Overview of the Worldline Approach to Semi-Classical Conformal Blocks

The paper entitled "Worldline approach to semi-classical conformal blocks" by Eliot Hijano, Per Kraus, and River Snively presents a comprehensive investigation into the semi-classical limits of conformal blocks in two-dimensional conformal field theory (2D CFT) with respect to their connection to three-dimensional (3D) gravity through the AdS/CFT correspondence. This paper specifically explores the use of particle worldline techniques in 3D geometries, such as conical defects and BTZ black holes, to understand the behavior of four-point functions with heavy and light operator exchanges.

Semi-Classical Conformal Blocks and AdS/CFT Correspondence

In 2D CFTs, conformal blocks are integral components for decomposing correlation functions, and are determined by conformal invariance. The semi-classical limit, which involves large central charge and operator dimensions while maintaining their ratios, is crucial for understanding how conformal blocks exponentiate, a behavior supported by significant evidence but without direct proof. The paper extends established knowledge by leveraging the monodromy method and worldline approach, providing a more accessible computation strategy by utilizing semiclassical geometries in the AdS context.

Primary Results

The authors successfully calculate the semi-classical conformal blocks through both monodromy methods and worldline configurations. The worldline approach simplifies the computation by depicting particle geodesics interconnected at cubic vertices within asymptotically AdS backgrounds. Their computations show agreement between the conformal field theory predictions and bulk gravity interpretations, specifically aligning results derived from CFT recursion relations and bulk geodesic probes. One of the paper's substantial findings is its analysis and extension of geodesic configurations involving multiple light operator exchanges, offering a compact yet complex formula for non-vacuum conformal blocks.

Implications and Future Directions

The implications of this paper primarily reside in its contribution to a more detailed picture of the AdS/CFT duality, enabling deeper insights into how local physics emerges from holographic principles. The framework proposed could guide future research into more intricate configurations, such as higher-point functions and computations on Riemann surfaces of greater genus. Additionally, the bulk interpretation indicates pathways for resolving aspects of information loss and chaos in AdS/CFT scenarios, pointing towards advancements in understanding thermal systems and entanglement entropy within a holographic scope.

Future developments may focus on expanding this approach to accommodate corrections beyond the semi-classical regime (1/c effects) or adapting these formulations to higher-dimensional or higher-spin systems. Such extensions would entail exploring W-algebras in addition to Virasoro algebras, further enriching the AdS/CFT correspondence landscape.

Conclusion

Overall, the paper significantly enriches the understanding of conformal blocks within 2D CFT and their parallels in 3D AdS gravity. It offers rigorous calculations along with simplified models that provide foundational insights for theoretical and practical applications in the field of holography and quantum gravity. This research stands as a strong basis for further explorations into the interaction of CFTs and bulk gravity while expanding the technical toolkit available to researchers in these domains.