Finite-size Effects from Giant Magnons

TL;DR

This paper analyzes finite-size effects for gauge-fixed strings on ${\rm AdS}_5\times {\rm S}^5$ by constructing one-soliton (giant magnon) configurations with finite charge $J$. It shows that at finite $J$ the magnon is gauge-dependent, the level-matching condition is violated, and the symmetry algebra breaks down from ${\mathfrak{psu}}(2,2|4)$, while the dispersion relation acquires exponential finite-size corrections with an effective length $\mathcal{R}$. In the infinite-$J$ limit, the configurations recover Hofman-Maldacena magnons with gauge-independent energy, and the authors extend the construction to two-spin magnons and discuss implications for Bethe-ansatz descriptions of strings with finite charges. The work also connects finite-$J$ results across uniform and conformal gauges and provides explicit, elliptic-function-based expressions for the finite-size corrections, offering a nontrivial check for any proposed finite-$J$ string Bethe ansatz.

Abstract

In order to analyze finite-size effects for the gauge-fixed string sigma model on AdS_5 x S^5, we construct one-soliton solutions carrying finite angular momentum J. In the infinite J limit the solutions reduce to the recently constructed one-magnon configuration of Hofman and Maldacena. The solutions do not satisfy the level-matching condition and hence exhibit a dependence on the gauge choice, which however disappears as the size J is taken to infinity. Interestingly, the solutions do not conserve all the global charges of the psu(2,2|4) algebra of the sigma model, implying that the symmetry algebra of the gauge-fixed string sigma model is different from psu(2,2|4) for finite J, once one gives up the level-matching condition. The magnon dispersion relation exhibits exponential corrections with respect to the infinite J solution. We also find a generalisation of our one-magnon configuration to a solution carrying two charges on the sphere. We comment on the possible implications of our findings for the existence of the Bethe ansatz describing the spectrum of strings carrying finite charges.

Figures (12)

• Figure 1: There are potentally two ways to take the limit from a finite $J$, two soliton configuration. One way is to have the solitons on "different sides" of the string: this leads to two one-soliton configurations, living on different lines. Another way is to have the solitons on the same "side" of the string: this leads to a nontrivial two-soliton configuration on the line. In the target space, the former configuration corresponds to a folded string with the shape of a giant magnon, which is a legitimate closed string state. In the latter case, sending $J$ to infinity, does not naturally opens up the string, since solitons remain unseparated in the limit. Only if the total worldsheet momentum is nonzero, the latter becomes a complicated open string state, which is such that when the total worldsheet momenta of solitons becomes zero, one is back to the closed string.
• Figure 2: At finite $J$ the two-soliton configuration is complicated and never a trivial superposition of two one-magnon solutions. This is the reason why we cannot trivially build a closed string state only from two magnons. At infinite $J$ the situation is different, and there is a trivial configuration of two magnons (see the upper right-hand side picture of figure \ref{['limi']}).
• Figure 3: Snapshots of the time evolution of the solution in conformal gauge.
• Figure 4: Profile of $a=0$ one-magnon soliton: Left, $z(\sigma)$ plotted for configurations with the same $z_{\text{max}}=0.99$ and $z_{\text{min}}= \{0.6,0.19,0.06 \}$, green, red and blue respectively. Right, profile $x^-(\sigma)$ for the same values of $z_{\text{min}}, z_{\text{max}}$.
• Figure 5: Target space shape of magnon at fixed light-cone time $x^{+}$, depicted for three magnons moving in the stripe $z_{\text{max}} = 0.99$, $z_{\text{min}} = \{0.6,0.19,0.06 \}$, green, red and blue respectively.