Exact Correlators on the Wilson Loop in $\mathcal{N}=4$ SYM: Localization, Defect CFT, and Integrability

TL;DR

This paper computes exact correlation functions of operator insertions on the 1/8 BPS Wilson loop in N=4 SYM by combining localization, OPE, and Gram–Schmidt orthogonalization. It reveals a determinant structure and a topological subsector of correlators, encodes higher-point data through polynomials F_L and D_L, and yields a large-N integral representation in terms of Q_L(x) tied to the Quantum Spectral Curve. The results specialize to the 1/2 BPS loop to provide exact defect CFT data, and show precise strong-coupling agreement with AdS_2 string theory as well as finite-N generalizations of generalized Bremsstrahlung functions. At large N, the correlators are recast as simple integrals over polynomials (Q-functions), highlighting a deep link between localization, defect CFT, and integrability, and suggesting fertile ground for bootstrap and hexagon-based approaches. Overall, the work unifies localization, integrability, and holography in a novel defect-CFT context and opens avenues for extending to broader operator content and other theories.

Abstract

We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in $\mathcal{N}=4$ SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the 't Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbitrary length inserted on the loop. At strong coupling, we show precise agreement with a direct calculation using perturbation theory around the AdS$_2$ string worldsheet. We also explain the connection of our results to the "generalized Bremsstrahlung functions" previously computed from integrability techniques, reproducing the known results in the planar limit as well as obtaining some of their finite N generalizations. Furthermore, we show that the correlators at large N can be recast as simple integrals of products of polynomials (known as Q-functions) that appear in the Quantum Spectral Curve approach. This suggests an interesting interplay between localization, defect CFT and integrability.

Figures (7)

• Figure 1: General configuration of the $1/8$-BPS Wilson loop, denoted by a red curve. The $1/8$-BPS Wilson loop lives on $S^2$ and couples to a scalar as prescribed in (\ref{['eq:defofWL']}). The expectation value of such a loop depends only on the area $A$ of the region inside the loop (the red-shaded region in the figure). Note that, although "the region inside/outside the loop" is not a well-defined notion, such ambiguity does not affect the expectation value since it is invariant under $A\to 4\pi -A$, which exchanges the regions inside and outside the loop.
• Figure 2: Cusped Wilson line with insertions. The cusped Wilson line consists of two semi-infinite lines which intersects with an angle $\phi$ at the origin, and the insertions $Z^{L}$. The scalar coupling of each semi-infinite line is given by the vector $\vec{n}_{1,2}$, and the relative angle between the two vectors is $\theta$. The divergence from this Wilson line is controlled by the generalized Bremsstrahlung function.
• Figure 3: Cusped Wilson loop on $S^2$. Applying the conformal transformation, one can map the cusped Wilson line to a configuration depicted above. The red and black semi-circles correspond to the two semi-infinite lines in figure \ref{['fig:fig1']} of the same color. The angle between the two semi-circles is $\pi-\theta$. The loop divides the $S^2$ into two regions with areas $2\pi \mp 2\theta$. (Note that we already set $\phi=\theta$ in this figure.)
• Figure 4: Witten diagrams in AdS$_2$ contributing to the 4-point function of single-letter insertions $\Phi$ to next-to-leading order at strong coupling. The grey blob in the middle figure denote the one-loop correction to the "boundary-to-boundary" $y$ propagator.
• Figure 5: Topology of Witten diagrams contributing to the 2-point function of $\Phi^L$ (in the picture the case $L=4$ is shown). The diagrams on the left, corresponding to generalized free-field contractions, also receive a subleading correction where a $y$-propagator is one-loop corrected.