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The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines

Michael Cooke, Amit Dekel, Nadav Drukker

TL;DR

The paper develops a defect CFT framework for operator insertions on the 1/2-BPS Wilson line in ${\mathcal{N}}=4$ SYM, establishing the precise mapping between small deformations of the line and local operator insertions. By expanding known weak-coupling Wilson-loop results to order ${\epsilon^4}$ and organizing insertions into $OSp(2,2|4)$ multiplets, the authors extract two-point coefficients, anomalous dimensions, and tree-level structure constants for low-dimension operators, including explicit results for ${\Phi}^1$, ${i\mathbb{F}}_{i3}$, and related multiplets. They show that certain operators are protected (vanishing anomalous dimension) and reveal the universal behavior of the first insertion through the Bremsstrahlung function $B(\lambda)$, connecting weak- and strong-coupling physics. The work also discusses regularization schemes, consistency with multiplet descent relations, and the prospects for extending the analysis to higher dimensions, more complicated insertions, or strong coupling via AdS/CFT, where the four-point function structure is not universally restricted. Overall, the results provide a concrete set of defect-CFT data for insertions on the Wilson line and a methodology to relate line deformations to operator data, with implications for integrability and holography.

Abstract

We study operator insertions into the $1/2$ BPS Wilson loop in ${\cal N}=4$ SYM theory and determine their two-point coefficients, anomalous dimensions and structure constants. The calculation is done for the first few lowest dimension insertions and relies on known results for the expectation value of a smooth Wilson loop. In addition to the particular coefficients that we calculate, our study elucidates the connection between deformations of the line and operator insertions and between the vacuum expectation value of the line and the CFT data of the insertions.

The Wilson loop CFT: Insertion dimensions and structure constants from wavy lines

TL;DR

The paper develops a defect CFT framework for operator insertions on the 1/2-BPS Wilson line in SYM, establishing the precise mapping between small deformations of the line and local operator insertions. By expanding known weak-coupling Wilson-loop results to order and organizing insertions into multiplets, the authors extract two-point coefficients, anomalous dimensions, and tree-level structure constants for low-dimension operators, including explicit results for , , and related multiplets. They show that certain operators are protected (vanishing anomalous dimension) and reveal the universal behavior of the first insertion through the Bremsstrahlung function , connecting weak- and strong-coupling physics. The work also discusses regularization schemes, consistency with multiplet descent relations, and the prospects for extending the analysis to higher dimensions, more complicated insertions, or strong coupling via AdS/CFT, where the four-point function structure is not universally restricted. Overall, the results provide a concrete set of defect-CFT data for insertions on the Wilson line and a methodology to relate line deformations to operator data, with implications for integrability and holography.

Abstract

We study operator insertions into the BPS Wilson loop in SYM theory and determine their two-point coefficients, anomalous dimensions and structure constants. The calculation is done for the first few lowest dimension insertions and relies on known results for the expectation value of a smooth Wilson loop. In addition to the particular coefficients that we calculate, our study elucidates the connection between deformations of the line and operator insertions and between the vacuum expectation value of the line and the CFT data of the insertions.

Paper Structure

This paper contains 20 sections, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Equivalence of operator insertions and deformations
  3. The operator expansion
  4. Deformations of the straight line and explicit form of the operators
  5. Operators representation and multiplets
  6. Operators of classical dimension one
  7. Operators of classical dimension two
  8. Operators of classical dimension three and four
  9. Extracting the CFT data
  10. The one and two-loop expectation values for general contours
  11. Divergences and regularization
  12. Order $\epsilon^2$
  13. One-loop
  14. Two-loop
  15. Order $\epsilon^3$
  16. ...and 5 more sections

Figures (2)

  • Figure 1: A Feynman diagram contributing to $\mathop{\mathrm{\langle\space\langle\space}}\nolimits{\mathcal{O}}_1{\mathcal{O}}_2\mathop{\mathrm{\space\rangle\space\rangle}}\nolimits$. The straight line represents the $1/2$ BPS line. The wavy lines represent propagators in the theory, the exact form of which depends on the operators ${\mathcal{O}}_1$ and ${\mathcal{O}}_2$.
  • Figure 2: The insertions of lowest dimensions with their quantum numbers under $SO(1,2)^2\times Sp(4)$ (the first is the dimension). The diagram shows the supermultiplets starting with the primary fields $\Phi^1$ and $\Phi^A$ and acting with the unbroken supercharges $Q^+_{\alpha,a}$ to generate the super-descendants. The dashed arrows show the action of the broken supercharges $Q^-_{\alpha,a}$, which take us from one supermultiplet to another.