Giant Magnons

TL;DR

The paper identifies giant magnons in AdS5×S5 as fundamental stringy excitations corresponding to spin-chain impurities in N=4 SYM and shows their dispersion relation arises from interpreting momentum as a geometric angle on the string. It demonstrates the matching SU(2|2)×SU(2|2) symmetry and central charges on both gauge and string sides, and computes the semiclassical S-matrix at large coupling, finding agreement with earlier string Bethe results. A sine-Gordon correspondence underpins the dispersion and scattering analysis, revealing an infinite tower of two-magnon bound states at strong coupling. Overall, the work solidifies the integrable structure of the AdS/CFT system, linking gauge theory dynamics with classical and semiclassical string theory via exact S-matrix constraints and bound-state spectra.

Abstract

Studies of ${\cal N}=4$ super Yang Mills operators with large R-charge have shown that, in the planar limit, the problem of computing their dimensions can be viewed as a certain spin chain. These spin chains have fundamental ``magnon'' excitations which obey a dispersion relation that is periodic in the momentum of the magnons. This result for the dispersion relation was also shown to hold at arbitrary 't Hooft coupling. Here we identify these magnons on the string theory side and we show how to reconcile a periodic dispersion relation with the continuum worldsheet description. The crucial idea is that the momentum is interpreted in the string theory side as a certain geometrical angle. We use these results to compute the energy of a spinning string. We also show that the symmetries that determine the dispersion relation and that constrain the S-matrix are the same in the gauge theory and the string theory. We compute the overall S-matrix at large 't Hooft coupling using the string description and we find that it agrees with an earlier conjecture. We also find an infinite number of two magnon bound states at strong coupling, while at weak coupling this number is finite.

Figures (9)

• Figure 1: Localized excitations propagating along the flat space string worldsheet in light cone gauge. (a) Worldsheet picture of the light cone ground state, with $P_+=0$. (b) Worldsheet picture of two localized excitations with opposite momenta propagating along the string. (c) Spacetime description of the configurations in (a) and (b). The configuration in (a) gives a straight line at a constant $X^-$. The configuration in (b) gives two straight lines at constant $X^-$ when the localized excitations are separated on the worldsheet. When the two excitations in (b) cross each other the lines move in $X^-$. (d) Snapshot of the spacetime configuration in (b), (c) at a given time $t$.
• Figure 2: Localized excitations propagating on an infinite string. (a) Worldsheet picture of a localized excitation propagating along the string. (b) Spacetime behavior of the state in lightcone coordinates. We have two lightlike lines with a string stretching between them. (c) Snapshot of the state at a given time. The configuration moves to the right at the speed of light.
• Figure 3: Giant magnon solution to the classical equations. The momentum of the state is given by the angular distance between the endpoints of the string. We depicted a configuration with $0 < p < \pi$. A configuration with negative momentum would look the same except that the orientation of the string would be reversed. The string endpoints are on the equator and the move at the speed of light.
• Figure 4: Giant magnons in LLM coordinates (\ref{['llmcoord']}). (a) A giant magnon solution looks like a straight stretched string. It's momentum $p$ is the angle subtended on the circle. $k_1$ and $k_2$ are the projections of the string along the directions $\hat{1}$ and $\hat{2}$. The direction of the string gives the phase of $k_1 + i k_2$, while its length gives the absolute value of the same quantity. (b) A closed string state built of magnons that are well separated on the worldsheet. Notice that the total central charges $k_1,~ k_2$ vanish. Similarly the total angle subtended by the string, which is the total momentum $p_{total}$ also vanishes modulo $2 \pi$.
• Figure 5: Spinning string configuration that corresponds to two magnons with $p= \pi$.