Perturbative bootstrap of the Wilson-line defect CFT: Bulk-defect-defect correlators

TL;DR

This work develops a perturbative bootstrap program for bulk-defect-defect correlators in the Wilson-line defect CFT of ${\mathcal N}=4$ SYM, focusing on one bulk and two defect half-BPS operators. By combining non-perturbative constraints from superconformal symmetry and localization with explicit weak- and strong-coupling computations, the authors determine the structure of these correlators and extract new OPE data, including higher-trace defect contributions. A striking outcome at weak coupling is the systematic cancellation of transcendental terms at next-to-leading order, explained by SUSY-related relations among OPE coefficients and scaling dimensions. At strong coupling, the leading and next-to-leading contributions to the correlator ${\langle 2\hat{1}\hat{1} \rangle}$ are obtained from a mix of Witten diagrams and non-perturbative constraints, yielding logarithmic terms and new CFT data that include higher-trace effects, and highlighting the role of AdS$_2$ dynamics in defect CFTs.

Abstract

We study the correlators of bulk and defect half-BPS operators in $\mathcal{N}=4$ Super Yang-Mills theory with a Maldacena-Wilson line defect, focusing on the case involving one bulk and two defect local operators. We analyze the non-perturbative constraints on these correlators, which include a topological sector, pinching and splitting limits, and we compute a variety of bulk-defect-defect correlators up to next-to-leading order at weak coupling, surprisingly observing that transcendental terms cancel. Additionally, we provide results in the strong-coupling regime for the first two leading orders using a mixture of Witten diagrams and non-perturbative constraints.

Figures (7)

• Figure 1: Illustration of a bulk-defect-defect correlator $\langle\, \mathcal{O}_1 \hat{\mathcal{O}}_2 \hat{\mathcal{O}}_3 \, \rangle$. The two defect operators $\hat{\mathcal{O}}_{2,3}$ are representations of the one-dimensional CFT preserved by the line, while the bulk operator $\mathcal{O}_1$ on the right lives in the four-dimensional space of ${\mathcal{N}}=4$ SYM.
• Figure 2: Illustration of the operator product expansion for the bulk-defect-defect correlators. The result is an expansion in conformal blocks, with the OPE coefficients being given by the product of defect three-point and bulk-defect two-point coefficients.
• Figure 3: The left figure displays the topological sector for $\langle\, 2 \hat{1} \hat{1} \, \rangle$. The blue line is the exact expression, presented in \ref{['eq:Fds211_Exact']}. The green and orange lines correspond, respectively, to the weak- and strong-coupling expansions up to next-to-leading order, derived in Giombi:2018hsx and reproduced here in \ref{['eq:Fds_weak']}, \ref{['eq:Fds1_weak']} and \ref{['eq:Fds211_strong']}. On the right, the analytic structure of the superblock ${\mathcal{G}}_{{\hat{\Delta}} =2} (\zeta;x)$ is presented. We see a discontinuity at $x \geq 1$, which is incompatible with the locality requirement discussed in Levine:2023ywqLevine:2024wqn. Note that the superblocks ${\mathcal{G}}_{{\hat{\Delta}} > 2} (\zeta;x)$ exhibit the same behavior.
• Figure 4: Analytic structure of the correlator ${\mathcal{A}}_{2 \hat{1} \hat{1}}$ at next-to-leading order at weak coupling. The left figure shows the $R$-symmetry channel $F_1(x)$, while the second one corresponds to $F_2(x)$. We observe that the correlator is discontinuous at $x \leq 0$, while it is continuous at $x=1$. This matches perfectly the analysis of Levine:2023ywqLevine:2024wqn.
• Figure 5: Complex plot of the Feynman diagram given in Equation \ref{['eq:NLO_FeynmanDiagrams_F2_211_1']} Here, we defined $f(x) = \bigl(\pi + \arctan (\sqrt{(1-x)/x}) \bigr)^2$. We observe a discontinuity at $x \geq 0$, incompatible with the locality constraint.