Half-BPS Wilson loop and AdS$_2$/CFT$_1$

TL;DR

The paper develops the AdS$_2$/CFT$_1$ description of operator insertions along a 1/2-BPS Wilson line in ${\cal N}=4$ SYM, identifying light AdS$_2$ fluctuations $y^a$ and $x^i$ as dual to protected dimension operators with $\Delta=1$ and $\Delta=2$ in the defect CFT$_1$. By expanding the AdS$_5\times S^5$ string action around the straight-line minimal surface and computing tree-level AdS$_2$ Witten diagrams, it derives strong-coupling four-point functions for insertions and extracts leading anomalous dimensions of two-particle operators via the OPE, including explicit results for $S^5$ fluctuations and mixed AdS$_5$/$S^5 fluctuations. The analysis reveals universal large-$n$ behavior of two-particle operator dimensions, highlights operator mixing in several channels, and provides precise OPE coefficient corrections at order $1/\sqrt{\lambda}$. For the circular Wilson loop, the AdS$_2$ results reconcile with localization predictions from 2d YM theory, offering a nontrivial consistency check of the AdS$_2$/CFT$_1$ framework and illustrating the connection between defect correlators and integrability data. $

Abstract

We study correlation functions of local operator insertions on the 1/2-BPS Wilson line in ${\cal N}=4$ super Yang-Mills theory. These correlation functions are constrained by the 1d superconformal symmetry preserved by the 1/2-BPS Wilson line and define a defect CFT$_1$ living on the line. At strong coupling, a set of elementary operator insertions with protected scaling dimensions correspond to fluctuations of the dual fundamental string in AdS$_5 \times S^5$ ending on the line at the boundary and can be thought of as light fields propagating on the AdS$_2$ worldsheet. We use AdS/CFT techniques to compute the tree-level AdS$_2$ Witten diagrams describing the strong coupling limit of the four-point functions of the dual operator insertions. Using the OPE, we also extract the leading strong coupling corrections to the anomalous dimensions of the "two-particle" operators built out of elementary excitations. In the case of the circular Wilson loop, we match our results for the 4-point functions of a special type of scalar insertions to the prediction of localization to 2d Yang-Mills theory.

Figures (2)

• Figure 1: Four-point function of local operators inserted on the Wilson line from a Witten diagram on the AdS$_2$ worldsheet.
• Figure 2: The disconnected and connected contributions to the 4-point function.