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

TL;DR

This work develops a perturbative bootstrap approach for multipoint defect CFT data on the Maldacena-Wilson line in ${ m N}=4$ SYM at large $N$ and weak coupling. By combining non-perturbative constraints from superconformal symmetry, crossing, and pinching with a single pivotal input—the six-point train track integral—the authors determine five- and six-point defect correlators at next-to-next-to-leading order, and provide new analytic results for certain four-point functions. The method reduces the dynamical content to one or two cross-ratio functions, enabling a largely diagrammatic yet highly constrained determination of correlators. These results deepen the bootstrap- and integrability-inspired understanding of higher-point data in defect CFTs and point toward recursive strategies for even higher-point functions and connections to localization and fishnet-type theories.

Abstract

We study the defect CFT associated with the half-BPS Wilson line in $\mathcal{N}=4$ Super Yang-Mills theory in four dimensions. Using a perturbative bootstrap approach, we derive new analytical results for multipoint correlators of protected defect operators at large $N$ and weak coupling. At next-to-next-to-leading order, we demonstrate that the simplest five- and six-point functions are fully determined by non-perturbative constraints -- which include superconformal symmetry, crossing symmetry, and the pinching of operators to lower-point functions -- as well as by a single integral, known as the train track integral. Additionally, we present new analytical results for the four-point functions $\langle 1122 \rangle$ and $\langle 1212 \rangle$.

Figures (5)

• Figure 1: Illustration of the strategy used for calculating multipoint correlation functions. The center column consists of the correlators that are computed throughout this paper. The left column refers to the data accessible non-perturbatively, while the right column refers to the perturbative input. The correlators $\langle\, 1111 \, \rangle$ and $\langle\, 1122 \, \rangle$ require the input of one $R$-channel each, while the other correlators necessitate the topological data ${\mathds{F}}$ and a single six-point integral, known as the train track.
• Figure 2: Illustration of the multipoint correlators defined in \ref{['eq:DefinitionCorrelators']}. The half-BPS defect operators are ordered along the Wilson line.
• Figure 3: Illustration of the steps involved in the bootstrap method used to derive the correlator $\langle\, 11112 \, \rangle$ at next-to-next-to-leading order. Superconformal Ward identities and crossing symmetry determine five of the six $R$-channels, in the basis $\tilde{R}_j$. An educated Ansatz is then formulated based on homogeneous transcendentality, limiting the number of open coefficients to $231$. By inputting the train track integral \ref{['eq:TrainTrack_Result']}, which completely determines the channel $F_1$, all but $20$ coefficients are fixed. The remaining coefficients are resolved using the pinching constraints discussed in Section \ref{['subsubsec:Pinching_FivePointFunctions']}.
• Figure 4: Illustration of the method used for bootstrapping the six-point function $\langle\, 111111 \, \rangle$ at next-to-next-to-leading order.
• Figure 5: The two OPE configurations used for performing checks on our result for the six-point function $\langle\, 111111 \, \rangle$ are displayed here. The left figure presents the comb channel and the leading non-trivial exchange of operators. The right figure shows the leading non trivial contribution in the snowflake channel.