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Spinning operators and defects in conformal field theory

Edoardo Lauria, Marco Meineri, Emilio Trevisani

TL;DR

<3-5 sentence high-level summary>This work develops a comprehensive framework for spinning operators in conformal field theories with defects, introducing a minimal, embedding-space construction of mixed-symmetry tensor structures via polynomials and group-theory projectors. It then builds the full spinning conformal blocks for two-point functions in both bulk and defect channels, using radial expansions, Zamolodchikov-type recurrences, and differential operators (including spin-transfer operators) to generate higher-spin blocks from seed blocks. The defect channel, in particular, admits a closed-form seed-block description through $SO(p+1,1) imes SO(q)$ projectors, with a complete set of spinning blocks obtained by systematic operator action and recurrence relations. The scalar Wilson line example validates the framework and demonstrates concrete OPE data and block decompositions, illustrating the practical utility for studying defects in higher-dimensional CFTs.

Abstract

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 operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.

Spinning operators and defects in conformal field theory

TL;DR

<3-5 sentence high-level summary>This work develops a comprehensive framework for spinning operators in conformal field theories with defects, introducing a minimal, embedding-space construction of mixed-symmetry tensor structures via polynomials and group-theory projectors. It then builds the full spinning conformal blocks for two-point functions in both bulk and defect channels, using radial expansions, Zamolodchikov-type recurrences, and differential operators (including spin-transfer operators) to generate higher-spin blocks from seed blocks. The defect channel, in particular, admits a closed-form seed-block description through projectors, with a complete set of spinning blocks obtained by systematic operator action and recurrence relations. The scalar Wilson line example validates the framework and demonstrates concrete OPE data and block decompositions, illustrating the practical utility for studying defects in higher-dimensional CFTs.

Abstract

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 operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.

Paper Structure

This paper contains 61 sections, 251 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mixed symmetry representations and CFTs
  3. Tensor structures for $SO(n)$ mixed symmetry representations
  4. Mixed symmetry tensors as polynomials
  5. Projectors onto representations of $SO(n)$
  6. Tensor structures for $SO(d+1,1)$ mixed symmetry representations
  7. Examples of correlation functions
  8. Two-point functions
  9. Three-point functions
  10. Mixed symmetry representations and defect CFTs
  11. Operators and tensor structures
  12. Examples of correlation functions
  13. One-point functions
  14. Two-point functions of bulk traceless and symmetric operators
  15. Two-point functions of generic defect operators
  16. ...and 46 more sections

Figures (1)

  • Figure 1: The bulk and defect radial frames, corresponding to equations \ref{['B_Poin_Conf']} and \ref{['D_Poin_Conf']}. Bulk radial frame (left): the defect is spherical and orthogonal to the plane drawn in the figure, and crosses it at the position marked by the red dots. The operators $\mathcal{O}_1$ and $\mathcal{O}_2$ sit at the same radius $r$. Defect radial frame (right): the defect is flat and orthogonal to the plane drawn in the figure, and crosses it at the position marked by the red dot. The operator $\mathcal{O}_1$ sits at unit radius, while $\mathcal{O}_2$ lies at radius ${\hat{r}}$.