# Recursion relations for conformal blocks

@article{Penedones2015RecursionRF, title={Recursion relations for conformal blocks}, author={Jo{\~a}o Penedones and Emilio Trevisani and Masahito Yamazaki}, journal={Journal of High Energy Physics}, year={2015}, volume={2016}, pages={1-51} }

A bstractIn the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension Δ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the…

## 147 Citations

### Recursion relations for 5-point conformal blocks

- Mathematics, PhysicsJournal of High Energy Physics
- 2021

We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal…

### Weight shifting operators and conformal blocks

- Mathematics
- 2017

A bstractWe introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving…

### Radial expansion for spinning conformal blocks

- Mathematics
- 2016

A bstractThis paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the…

### Recursion relation for general 3d blocks

- Computer ScienceJournal of High Energy Physics
- 2019

We derive closed-form expressions for all ingredients of the Zamolodchikov-like recursion relation for general spinning conformal blocks in 3-dimensional conformal field theory. This result opens a…

### Weight shifting operators and conformal blocks

- MathematicsJournal of High Energy Physics
- 2018

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with…

### Multipoint conformal blocks in the comb channel

- MathematicsJournal of High Energy Physics
- 2019

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for n-point…

### Gaudin models and multipoint conformal blocks: general theory

- MathematicsJournal of High Energy Physics
- 2021

The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory.…

### Spinning operators and defects in conformal field theory

- MathematicsJournal of High Energy Physics
- 2019

We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary…

### Closed-form expression for cross-channel conformal blocks near the lightcone

- MathematicsJournal of High Energy Physics
- 2020

In the study of conformal field theories, conformal blocks in the lightcone limit are fundamental to the analytic conformal bootstrap method. Here we consider the lightcone limit of 4-point functions…

### Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory

- Mathematics
- 2017

Conformal blocks are the central ingredient of the conformal bootstrap programme. We elaborate on our recent observation that uncovered a relation with wave functions of an integrable…

## References

SHOWING 1-10 OF 59 REFERENCES

### Spinning conformal blocks

- Physics, Mathematics
- 2011

This paper introduces a method to derive the conformal blocks corresponding to the exchange of a traceless symmetric tensor appearing in four point functions of operators with spin using the embedding space formalism.

### Radial expansion for spinning conformal blocks

- Mathematics
- 2016

A bstractThis paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the…

### Radial Coordinates for Conformal Blocks

- Mathematics
- 2013

We develop the theory of conformal blocks in CFT_d expressing them as power series with Gegenbauer polynomial coefficients. Such series have a clear physical meaning when the conformal block is…

### Diagonal limit for conformal blocks in d dimensions

- Mathematics
- 2013

A bstractConformal blocks in any number of dimensions depend on two variables z, $ \overline{z} $. Here we study their restrictions to the special “diagonal” kinematics $ z=\overline{z} $, previously…

### Projectors, shadows, and conformal blocks

- Mathematics
- 2014

A bstractWe 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…

### Spinning conformal correlators

- Computer Science
- 2011

A bstractWe develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an indexfree notation where tensors…

### Bounds in 4D conformal field theories with global symmetry

- Mathematics, Computer Science
- 2010

The constraining power of OPE associativity in 4D conformal field theory with a continuous global symmetry group is explored and an upper bound on the dimension of the lowest singlet scalar in the ϕ × ϕ† OPE is proved.

### Conformal correlators of mixed-symmetry tensors

- Mathematics
- 2014

A bstractWe generalize the embedding formalism for conformal field theories to the case of general operators with mixed symmetry. The index-free notation encoding symmetric tensors as polynomials in…

### Deconstructing conformal blocks in 4D CFT

- Mathematics
- 2015

A bstractWe show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field…