Projectors, Shadows, and Conformal Blocks (1204.3894v6)

Abstract: We introduce a method for computing conformal blocks of operators in arbitrary Lorentz representations in any spacetime dimension, making it possible to apply bootstrap techniques to operators with spin. The key idea is to implement the "shadow formalism" of Ferrara, Gatto, Grillo, and Parisi in a setting where conformal invariance is manifest. Conformal blocks in $d$-dimensions can be expressed as integrals over the projective null-cone in the "embedding space" $\mathbb{R}^{d+1,1}$. Taking care with their analytic structure, these integrals can be evaluated in great generality, reducing the computation of conformal blocks to a bookkeeping exercise. To facilitate calculations in four-dimensional CFTs, we introduce techniques for writing down conformally-invariant correlators using auxiliary twistor variables, and demonstrate their use in some simple examples.

• The paper presents a method that computes conformal blocks for operators with arbitrary spin using the embedding space and shadow formalisms.
• It leverages conformal integrals simplified by Feynman parameterization and expresses results through elementary hypergeometric functions in even dimensions.
• The technique isolates conformal blocks from their shadows via monodromy projection and is effectively applied to higher spin computations in four-dimensional CFTs using twistors.

Analysis of "Projectors, Shadows, and Conformal Blocks"

The paper "Projectors, Shadows, and Conformal Blocks" by David Simmons-Duffin presents a method to compute conformal blocks of operators with arbitrary spin in any spacetime dimension, thereby extending the scope of conformal bootstrap techniques to include operators with spin. The paper leverages the "shadow formalism" introduced by Ferrara, Gatto, Grillo, and Parisi, ensuring conformal invariance throughout the calculations.

Key Contributions

1. Embedding Space Formalism: The paper begins by utilizing the embedding space formalism, which makes conformal invariance manifest by allowing operators to be expressed in terms of higher-dimensional flat space. This method simplifies the action of the conformal group and facilitates the construction of conformal integrals.
2. Shadow Formalism: At the core of the proposed method is the shadow formalism, which introduces shadow operators as nonlocal counterparts to conformal operators. These enable the computation of conformal blocks through integrals over the projective null-cone in the embedding space.
3. Computation of Conformal Integrals: The paper details computations of conformal integrals, which play a crucial role in evaluating conformal blocks. The approach involves the use of Feynman parameterization to simplify the integration process. The results are presented in terms of elementary hypergeometric functions in even spacetime dimensions.
4. Projection and Monodromy: A novel aspect of the work is the use of monodromy projection to isolate the conformal block associated with a specific conformal field from its shadow. This involves considering the analytic behavior of integrals under the transformation of cross-ratios.
5. Higher Spin Block Calculations: The method is successfully applied to compute higher spin conformal blocks in various scenarios, including four-dimensional conformal field theories (CFTs) using twistors. For instance, the exchange of an antisymmetric tensor in a four-point function of scalar and vector operators is explored.
6. Application to Four-Dimensional CFTs: The work extends to proposing a twistor formalism for four-dimensional CFTs, facilitating computations involving arbitrary Lorentz representations. Twistor variables offer a significant simplification in writing conformally invariant correlators.

Implications and Future Work

This work has profound implications for conformal bootstrap methods, potentially enabling a deeper exploration of theories where spinful operators play a critical role. By accommodating operators in arbitrary Lorentz representations and dimensions, this approach broadens the applicability of the conformal bootstrap, which is pivotal in exploring theoretical constraints across CFTs.

The techniques presented can unravel insights in several domains, including quantum gravity through the AdS/CFT correspondence, and might be particularly valuable in studying higher-dimensional superconformal field theories. While the current results focus on classical computation schemes, future extensions could include superconformal theories or utilize numerical techniques to capture more complex scenarios.

Overall, this paper provides essential tools for advancing theoretical investigations in high-energy physics, offering a robust methodology for handling intricate operator interactions typically encountered in CFT studies. Researchers interested in canonical quantization, operator algebras, and cross-dimensional symmetry properties will find these techniques particularly advantageous for their analyses.