Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks (1508.00501v2)

Abstract: We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram," which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

• The paper introduces geodesic Witten diagrams that replace complex bulk integrations with geodesic paths to compute conformal blocks.
• It verifies the method with explicit calculations for scalar and higher-spin exchanges, matching established CFT results.
• The approach streamlines holographic computations, offering practical insights for advancing research in quantum gravity and string theory.

Analyzing Witten Diagrams and Conformal Blocks in AdS/CFT

The paper "Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks" presents a novel method for analyzing the intricate relationship between Witten diagrams in Anti-de Sitter (AdS) space and conformal blocks in Conformal Field Theory (CFT). The authors address a longstanding ambiguity in the AdS/CFT correspondence by providing a geometric understanding of conformal blocks through what they term "geodesic Witten diagrams."

Key Contributions

The authors introduce the concept of geodesic Witten diagrams, which are modifications of the standard exchange Witten diagrams traditionally used to calculate holographic correlation functions in AdS/CFT. In these diagrams, the cubic vertices are integrated not over the entirety of AdS but restricted to geodesics connecting boundary points pertinent to the operators involved. This approach has several implications:

• Simplification of Calculations: The procedure eliminates the need for complex integrations over bulk vertices, thereby simplifying the calculations associated with Witten diagrams.
• Efficiency and Intuitiveness: By leveraging the intrinsic conformal symmetry and the geometric structure of AdS space, this method provides a streamlined and intuitive framework for understanding scalar and higher spin exchanges.
• Direct Connection to Conformal Partial Waves: The geodesic Witten diagrams directly compute conformal blocks, the fundamental building blocks in the decomposition of CFT correlation functions. This offers a clearer path to matching AdS computations with their CFT counterparts.

Methodology and Results

The authors focus specifically on four-point functions of scalar operators and extend their analysis to higher spin scenarios, such as the exchange of vector fields. They investigate the decomposition of correlation functions into conformal blocks, providing explicit calculations that demonstrate the simplicity and elegance of their approach.

1. Scalar Exchanges: The geodesic diagrams adhere to the expected spectrum of double-trace operators present in the generalized free field theories associated with classical AdS spacetimes. The authors verify that their geodesic method reproduces existing known results for scalar conformal blocks in multiple dimensions.
2. Higher Spin Exchanges: For vector and spin-2 exchanges, they develop corresponding geodesic Witten diagrams, showing that these are also consistent with the known polynomial structures of conformal blocks in various dimensions. They verify their findings using the method of conformal Casimir operators, ensuring consistency with well-established conformal symmetry principles.
3. Computational Efficacy: The paper explores expressions for Witten diagrams that would traditionally require complex asymptotic expansions, simplifying them into more tractable forms. This is accomplished through an efficient application of position space techniques, which bypass traditional Mellin space approaches.
4. Loop Extensions: While primarily addressing tree-level diagrams, the paper lays foundational concepts that could be extended to loop diagrams, a challenging frontier for AdS/CFT calculations.

Implications and Future Directions

This paper opens several new avenues for research within the AdS/CFT framework:

• Practical Calculations: The methodology can be applied to simplify the calculations required for the analysis of holographic correlation functions, potentially impacting quantum gravity and string theory.
• Generalization to Higher Orders: There is an opportunity to adapt these concepts to loop corrections and diagrams involving more complex operator exchanges.
• Exploration of New Bulk-Geodesic Relationships: Understanding how these geodesic structures relate to other non-perturbative phenomena in AdS/CFT remains a fertile ground for exploration.

By providing a more geometrically and computationally efficient framework, this paper contributes significantly to the theoretical toolkit available for holographic duality studies, and it sets the stage for further developments in this vibrant domain of theoretical physics.