Spiky strings, light-like Wilson loops and pp-wave anomaly

Abstract

We consider rigid rotating closed strings with spikes in AdS5 dual to certain higher twist operators in N=4 SYM theory. In the limit of large spin when the spikes reach the boundary of AdS5, the solutions with different numbers of spikes are related by conformal transformations, implying that their energy is determined by the same function of the `t Hooft coupling f(lambda) that appears in the anomalous dimension of twist 2 operators or in the cusp anomaly. In the limit when the number of spikes goes to infinity, we find an equivalent description in terms of a string moving in an AdS pp-wave background. From the boundary theory point of view, the corresponding description is based on the gauge theory living in a 4d pp-wave space. Then, considering a charge moving at the speed of light, or a null Wilson line, we find that the integrated energy momentum tensor has a logarithmic UV divergence whose coefficient we call the "pp-wave anomaly". The AdS/CFT correspondence implies that, for N=4 SYM, this pp-wave anomaly should have the same value as the cusp anomaly. We verify this at lowest order in SYM perturbation theory. As a side result of our string theory analysis, we find new open string solutions in the Poincare patch of the standard AdS space which end on a light-like Wilson line and also in two parallel light-like Wilson lines at the boundary.

Figures (2)

• Figure 1: Surface ending on two parallel light-like Wilson lines given by $t=x\pm1$ in ${AdS}_{5}$ space in Poincare coordinates.
• Figure 2: Surface ending on a single light-like line given by $x=t$. Here we plot the shape of the string $z(x)$ at different values of $t=200$, $400$, $600$, $800$, $1000$. The $x$-axis is the boundary where the string ends at $x=t$ (allowing to identify each curve), and its end-point we identify as a quark. In addition, there is a (rounded) spike coming out of the horizon at a point given by $x\simeq -t$ (at $x=-t$ we have $z=\infty$). The shape of the string at $t<0$ follows from the symmetry $(x,t)\rightarrow(-x,-t)$. Thus, the quark and the spike come together at $t<0$; at around $t=0$ the spike actually disappears and then reappears again on the other side moving away from the quark as shown in the figure. If we associate the spike coming out of the horizon with a gluon then this describes quark-gluon scattering but further analysis is needed in order to substantiate this interpretation.