Bootstrapping the half-BPS line defect

TL;DR

This work develops a defect bootstrap program for half-BPS line defects in 4d ${\mathcal N}=4$ theories, focusing on a 1d CFT with ${\rm OSP}(4^*|4)$ symmetry on the defect. By combining full and mixed correlators of displacement multiplets $\mathcal{D}_1$, $\mathcal{D}_2$ and their OPE data, the authors derive comprehensive crossing equations and implement semidefinite programming to bound conformal dimensions and OPE coefficients. A striking mixed-system island emerges, consistent with Wilson lines in planar ${\mathcal N}=4$ SYM at strong coupling, and analytic perturbation theory around the strong-coupling point reproduces key features of the numerics, including a sparsest spectrum and controlled corrections. The results illuminate the defect-bulk interplay, offer cross-checks with localization data, and point to future directions in coupling to the 4d bulk and exploring other supersymmetric defects. Overall, the paper provides a concrete, symmetry-driven bootstrap framework for line defects in highly supersymmetric gauge theories and identifies a potentially unique crossing-solution consistent with known strong-coupling behavior.

Abstract

We use modern bootstrap techniques to study half-BPS line defects in 4d N=4 superconformal theories. Specifically, we consider the 1d CFT with OSP(4*|4) superconformal symmetry living on such a defect. Our analysis is general and based only on symmetries, it includes however important examples like Wilson and 't Hooft lines in N=4 super Yang-Mills. We present several numerical bounds on OPE coefficients and conformal dimensions. Of particular interest is a numerical island obtained from a mixed correlator bootstrap that seems to imply a unique solution to crossing. The island is obtained if some assumptions about the spectrum are made, and is consistent with Wilson lines in planar N=4 super Yang-Mills at strong coupling. We further analyze the vicinity of the strong-coupling point by calculating perturbative corrections using analytic methods. This perturbative solution has the sparsest spectrum and is expected to saturate the numerical bounds, explaining some of the features of our numerical results.

Figures (14)

• Figure 1: The allowed region for $C_{1,1,2}$, $C_{2,2,2}$ from localization for classical groups $G$. Extremal points corresponding to free theories are marked by red points and the planar theory in the fundamental representation is marked by a dotted red curve. The U(1) theory at $(\sqrt{2},2\sqrt{2})$ (for any value of the coupling) has the same OPE coefficients as the strong coupling limit of any other case that we looked at. The notation for the theories is $G_{\mathcal{R}}$, where the representation $\mathcal{R}$ is given by its Dynkin labels.
• Figure 2: Exchanging the points $2$ and $4$ is a symmetry of the system.
• Figure 3: Left: Upper bounds on $\Delta_{[0,0]}$ as a function of $\Lambda^{-1}$. Several fits (linear while ignoring the first 4 points, quadratic and cubic) were done are are plotted in orange. Extrapolated to $\Lambda\rightarrow \infty$, they lead to $\Delta_{[0,0]}\leq 2.009, 2.007, 1.986$ respectively. Right: Bounds on the difference $\Delta'_{[0,0]}-\Delta_{[0,0]}$ between the conformal dimensions of the first two longs for a given first long with dimension $\Delta_{[0,0]}$. The plot was done for $\Lambda=20, 30, \ldots, 80$ and only the allowed region for $\Lambda=80$ was shaded. The left red dot denotes the analytic solution \ref{['eq: most general analytic solution']} for $\xi=-1$, while the right one corresponds to $\xi=1$. For the other values of $\xi$ we have $\Delta_{[0,0]}=1$ and $\Delta'_{[0,0]}=2$, which is too low to be interesting.
• Figure 4: Upper bounds for $\Delta_{[2,0]}$, $\Delta_{[0,2]}$, $\Delta_{[0,1]}$ as a function of $\Delta_{[0,0]}$ for $\Lambda=10,20,30,35, 40$. Gap structures coming from the analytic solutions \ref{['eq: most general analytic solution']} for special values of the parameters are shown with red crosses. Among them there is an analytic solution for which $\Delta_{[0,0]}=1$ and $\Delta_{[0,1]}=\infty$, which explains why the bound on $\Delta_{[0,1]}$ diverges for small $\Delta_{[0,0]}$. It seems plausible that for $\Delta_{[0,0]}=2$, the bounds would converge to the strong coupling values $\Delta_{[2,0]}=5$, $\Delta_{[2,0]}=4$, $\Delta_{[0,1]}=3$ for infinite $\Lambda$.
• Figure 5: Left: the bounds on $C_{1,1,2}^2$ as a function of $\Delta'_{[0,0]}$ if the semi-short $\mathcal{C}_{[2,0]}$ is present. Right: the bounds on $C_{1,1,\mathcal{C}_{[2,0]}}^2$ as a function of $\Delta'_{[0,0]}$. The numerics are done for $\Lambda=10,20,\ldots, 80$ and the allowed regions for $\Lambda=80$ are shaded in orange. Analytic solutions from \ref{['eq: most general analytic solution']} are marked in red. For the purpose of comparison, we overlay in light blue on the left the allowed region of figure \ref{['fig:DDBootstrapOPEshort']}. One must keep in mind that if $\mathcal{C}_{[2,0]}$ decouples, we can identify $\Delta_{[0,0]}'$ with $\Delta_{[0,0]}$ here, since we consider a single gap in the long spectrum.