A Rosetta Stone for Wilson Line Defects

TL;DR

This work investigates whether a smooth correspondence can be established between weak gauge-theory and strong-string descriptions for the supersymmetric Wilson line defect in planar ${\mathcal N}=4$ SYM. By constructing and refining partition functions that encode single- and two-particle states at zero and infinite coupling, the authors propose an alphabet map that predicts a doubling of the scaling dimensions, $\Delta_\infty = 2\,\Delta_0$, for these sectors. They test this proposal against nonperturbative spectral data obtained from the Quantum Spectral Curve, extending the curve to states with transverse spin and confirming the predicted doubling for both single-particle and two-particle singlet sectors. The results provide evidence for a smooth gauge-string map in a highly solvable setting and suggest practical avenues, such as double-sided Padé approximants, to interpolate intermediate-coupling spectra and inform bootstrap approaches. Overall, the Wilson line defect serves as a Rosetta Stone for connecting weak and strong coupling degrees of freedom in a controlled, integrable Defect CFT context.

Abstract

In this paper, we discuss the construction of a map between weak (gauge) and strong (string) coupling degrees of freedom for the supersymmetric Wilson line-defect in the planar N=4 Super-Yang-Mills. By analysing the Partition Functions at zero and infinite coupling, we propose a map from degrees of freedom capturing single- and singlet two-particle states at zero coupling to infinite coupling. This map predicts that the dimension of states in these particular sectors doubles as it goes from zero to infinite coupling. We test this prediction against the non-perturbative spectrum of insertions on the Wilson line obtained using integrability. In addition to already available integrability-based results, we obtain the non-perturbative scaling dimension of the simplest non-trivial operator with transverse spin about the Wilson line, thereby extending the Quantum Spectral Curve construction to such charged sectors.

Figures (6)

• Figure 1: Inserted operators $O_\text{inserted}$ in the Wilson line can be seen as a single-trace operator where one of single letters is a Wilson line (only one Wilson line letter is allowed).
• Figure 2: Single-particle spectrum. The first operator is the super-primary of $[0]_1^{(0,0)}$, whose dimension goes from $\Delta_0$=1 to $\Delta_{\infty}=2$. The second operator is the super-primary of $[2]_{2}^{(0,0)}$, which goes from $\Delta_{0}=2$ to $\Delta_{\infty}=4$.
• Figure 3: Twist-2 spectrum of singlet operators in $[0]_{\Delta}^{(0,0)}$ multiplets. We suppose that this spectrum is equivalent to the two-particle spectrum.
• Figure 4: Spectrum of $\Delta_2$ and $\Delta_3$ operators displayed together with their respective Padé approximations: the integrability data is shown in red, and the approximation in blue. The weak and strong perturbative data is shown with the dashed lines.
• Figure 5: The difference between the integrability spectrum $\Delta_2$ and $\Delta_3$ and their approximations.