Semiclassical Virasoro Blocks from AdS$_3$ Gravity (1508.04987v2)

Abstract: We present a unified framework for the holographic computation of Virasoro conformal blocks at large central charge. In particular, we provide bulk constructions that correctly reproduce all semiclassical Virasoro blocks that are known explicitly from conformal field theory computations. The results revolve around the use of geodesic Witten diagrams, recently introduced in arXiv:1508.00501, evaluated in locally AdS$_3$ geometries generated by backreaction of heavy operators. We also provide an alternative computation of the heavy-light semiclassical block -- in which two external operators become parametrically heavy -- as a certain scattering process involving higher spin gauge fields in AdS$_3$; this approach highlights the chiral nature of Virasoro blocks. These techniques may be systematically extended to compute corrections to these blocks and to interpolate amongst the different semiclassical regimes.

• The paper demonstrates calculating semiclassical Virasoro blocks from AdS3 gravity using geodesic Witten diagrams in a unified framework.
• Key results include a rigorous derivation for the heavy-light limit block using bulk scalar fields and extracting chiral blocks via higher-spin currents.
• The findings provide insights into AdS3/CFT duality and gravity's backreaction effects, potentially aiding understanding of quantum gravity.

Semiclassical Virasoro Blocks from $\text{AdS}_3$ Gravity

The paper "Semiclassical Virasoro Blocks from $\text{AdS}_3$ Gravity" presents a novel approach to calculating Virasoro conformal blocks within the framework of AdS/CFT correspondence. The authors demonstrate how Virasoro blocks, specifically in the semiclassical limit of large central charge, can be constructed from geodesic Witten diagrams in three-dimensional anti-de Sitter ($\text{AdS}_3$) space.

Framework and Methodology

The authors propose a unified framework for the holographic computation of Virasoro blocks, emphasizing the context of large central charge $c$. The methodology hinges on geodesic Witten diagrams, which are modified versions of traditional Witten diagrams where the integration over cubic vertices is constrained to geodesics connecting boundary points. This framework allows the recovery of all known semiclassical Virasoro blocks within certain limits by evaluating these diagrams in locally $\text{AdS}_3$ geometries featuring backreaction by heavy operators.

Among the various semiclassical regimes, the paper particularly highlights the heavy-light semiclassical block. In this regime, two external operators are significantly heavier compared to others, leading to a compelling result described as an interaction akin to a scattering process involving higher-spin gauge fields in $\text{AdS}_3$.

Key Results

1. Geodesic Witten Diagrams: The paper successfully demonstrates that geodesic Witten diagrams can compute semiclassical Virasoro blocks. These diagrams effectively translate the complex conformal block computations to more manageable geometric entities within the bulk $\text{AdS}_3$ spacetime.
2. Heavy-Light Limit: For the heavy-light limit, wherein certain operator dimensions grow linearly with the central charge $c$, the paper provides a rigorous derivation using bulk scalar fields. The result aligns with predictions from conformal field theory (CFT) and highlights the chiral nature of Virasoro blocks.
3. Higher Spin Currents: The paper explores the utility of higher spin currents to extract chiral blocks directly, supplementing the traditional scalar field approach. By leveraging the Chern-Simons formulation, researchers circumvent the complexities associated with symmetric traceless tensors, thus being able to directly compute the holomorphic conformal block.

Implications

The findings have profound implications on both theoretical and practical fronts. Theoretically, the work provides insights into the duality between gravitational interactions in $\text{AdS}_3$ and operator product expansions within CFTs. The established framework could serve as a foundation for exploring more sophisticated holographic computations involving complex conformal structures.

Practically, the framework is likely to enhance our understanding of gravitational backreaction effects in $\text{AdS}_3$, potentially illuminating aspects of quantum gravity linked to holography. The setup can be a stepping stone towards exploring quantum corrections in $\text{AdS}_3$, addressing nonperturbative aspects of quantum field theories and their gravitational duals.

Future Directions

The paper envisions extending the current formulations to compute further corrections in the conformal blocks, integrating $1/c$ quantum corrections, and expanding into nontrivial topological backgrounds. These developments could broaden the understanding of the emergence of bulk locality from CFTs and extend into exploring higher $W$ algebras in constrained systems. Moreover, this framework opens pathways to reconcile, or at least contrast, predictions of semiclassical gravity with quantum gravitational corrections, potentially leading to new discoveries in the landscape of theoretical physics.

Through rigorous discussion and innovative use of holographic principles, this paper contributes a significant step toward elucidating the deep connections linking gravity and conformal field theories.