On the integrability of Wilson loops in AdS_5 x S^5: Some periodic ansatze

TL;DR

This work demonstrates that the integrable structure of the $AdS_5\times S^5$ string sigma-model can be exploited to construct and analyze a wide class of Wilson-loop minimal surfaces. By employing periodic ans"atze inspired by spinning strings, the authors reduce the problem to Neumann-Rosochatius-type integrable systems in both the $S^5$ and $AdS_5$ sectors, solving many cases in terms of elliptic (and hyperelliptic) integrals. The paper provides explicit solutions in various subspaces ($AdS_2$, $AdS_3\times S^1$, $AdS_2\times S^2$, $AdS_3\times S^3$) and reveals a rich phase structure, including connected versus disconnected minimal surfaces and phase transitions, dependent on boundary data. These results illustrate the depth of the integrable approach for non-local observables in the AdS/CFT correspondence and raise important questions about how boundary conditions map to gauge-theory data and how the full planar theory should accommodate Wilson loops.

Abstract

Wilson loops are calculated within the AdS/CFT correspondence by finding a classical solution to the string equations of motion in AdS_5 x S^5 and evaluating its action. An important fact is that this sigma-model used to evaluate the Wilson loops is integrable, a feature that has gained relevance through the study of spinning strings carrying large quantum numbers and spin-chains. We apply the same techniques used to solve the equations for spinning strings to find the minimal surfaces describing a wide class of Wilson loops. We focus on different cases with periodic boundary conditions on the AdS_5 and S^5 factors and find a rich array of solutions. We examine the different phases that appear in the problem and comment on the applicability of integrability to the general problem.

Figures (8)

• Figure 1: The value of $\hat{y}$ as a function of $\hat{r}_1$ for three solutions to the $AdS_3$ ansatz. For two circles with radii $R_i=0.6$ and $R_f=1$ and opposite orientation (solid line), the solution has $a^2+p^2>0$. If the circles have the same orientation, the surface has to cross $\hat{r}_1=0$ which is given by the expressions with $a^2+p^2<0$ (dotted line). For $a^2+p^2=0$ (dashed line) the surface describes the correlator of a circle and a local operator at $\hat{r}_1=0$.
• Figure 2: The allowed range of $R_f/R_i$ for two circles of radii $R_f\geq R_i$ as a function of the separation on the sphere, $\delta\varphi_1$. Connected classical solutions exist for all values inside the region bound by the dashed line. This connected solution dominates only inside the region bound by the solid line.
• Figure 3: Phase diagram for two coincident circles with rotation on $S^2$ at angles $\theta_i+\theta_f=\pi$. Regular connected solutions exist only to the left of the dashed line, when $\theta_f-\theta_i$ is not too small, and $m/k$ not too large. The disconnected solution has smaller action in the entire range except for the small region above the solid line.
• Figure 4: The values of $\theta$ as a function of the spatial coordinate $x_2$ for boundary values $\theta_i=5\pi/12$ and $\theta_f=2\pi/3$ (the dotted horizontal lines) and different separation. At very large separation ($mR=3,\,4$) there are connected solutions (dashed) but the disconnected solution (solid) dominates. As shorter separation ($mR=1.5,\,2,\,2.5$) the connected solutions dominate and have a turning point $\theta_m>\theta_f$. At shorter distances the solution does not have a turning point.
• Figure 5: $\theta_m$, the minimal value of $\theta$ for two lines separated a distance $R$ with wrapping $m$ times around the sphere. The initial and final values of $\theta$ (which are equal in this case) can be read from the intersect of the lines with the axis at $R=0$, since then $\theta_m=\theta_i$. The bend in the curve gets sharper the larger $\sin\theta_i$ is, and for $\theta_i=\pi/2$ it is not differentiable.