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Bootstrap equations for $\mathcal{N}=4$ SYM with defects

Pedro Liendo, Carlo Meneghelli

Abstract

This paper focuses on the analysis of $4d$ $\mathcal{N}=4$ superconformal theories in the presence of a defect from the point of view of the conformal bootstrap. We will concentrate first on the case of codimension one, where the defect is a boundary that preserves half of the supersymmetry. After studying the constraints imposed by supersymmetry, we will obtain the Ward identities associated to two-point functions of $\tfrac{1}{2}$-BPS operators and write their solution as a superconformal block expansion. Due to a surprising connection between spacetime and R-symmetry conformal blocks, our results not only apply to $4d$ $\mathcal{N}=4$ superconformal theories with a boundary, but also to three more systems that have the same symmetry algebra: $4d$ $\mathcal{N}=4$ superconformal theories with a line defect, $3d$ $\mathcal{N}=4$ superconformal theories with no defect, and $OSP(4^*|4)$ superconformal quantum mechanics. The superconformal algebra implies that all these systems possess a closed subsector of operators in which the bootstrap equations become polynomial constraints on the CFT data. We derive these truncated equations and initiate the study of their solutions.

Bootstrap equations for $\mathcal{N}=4$ SYM with defects

Abstract

This paper focuses on the analysis of superconformal theories in the presence of a defect from the point of view of the conformal bootstrap. We will concentrate first on the case of codimension one, where the defect is a boundary that preserves half of the supersymmetry. After studying the constraints imposed by supersymmetry, we will obtain the Ward identities associated to two-point functions of -BPS operators and write their solution as a superconformal block expansion. Due to a surprising connection between spacetime and R-symmetry conformal blocks, our results not only apply to superconformal theories with a boundary, but also to three more systems that have the same symmetry algebra: superconformal theories with a line defect, superconformal theories with no defect, and superconformal quantum mechanics. The superconformal algebra implies that all these systems possess a closed subsector of operators in which the bootstrap equations become polynomial constraints on the CFT data. We derive these truncated equations and initiate the study of their solutions.

Paper Structure

This paper contains 67 sections, 154 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Correlation functions in superspace
  3. Superspace setup
  4. The supergroup $OSP(4|4)$.
  5. The superalgebra $\mathfrak{osp}(4|4)$.
  6. Superspace coordinates and their transformations.
  7. Superspace description of codimension two defects.
  8. Correlation functions
  9. Supermultiplets/Superfields.
  10. Covariance properties of correlation functions.
  11. One-point functions of bulk operators.
  12. Bulk-boundary two-point functions.
  13. Correlation functions in the presence of a $\frac{1}{2}$-BPS line defect and other examples
  14. $\frac{1}{2}$-BPS line defect.
  15. Correlators in $d=3$, $\mathcal{N}=4$ superconformal theories.
  16. ...and 52 more sections

Figures (2)

  • Figure 1: Crossing symmetry for two-point functions in the presence of a defect. The channel on the left represents the standard bulk OPE between local operators. The channel on the right is the bulk-to-defect OPE in which each local operator can be written as a convergent sum of defect operators.
  • Figure 2: On the left we have the configuration space for the spacetime coordinates and on the right for the R-symmetry coordinates. The subgroup of the conformal group unbroken in the presence of the defect is clear from the picture. On the left is the group of conformal transformations of the three-dimensional boundary $SO(3,2)$. On the right is the product of rotations $SO(3)$ in the space orthogonal to the line with the $SL(2,\mathbb{R})$ conformal transformations on the line.