Quantum wrapped giant magnon

Understanding the finite-size corrections to the fundamental excitations of a theory is the first step towards completely solving for the spectrum in finite volume. We compute the leading exponential correction to the quantum energy of the fundamental excitation of the light-cone gauged string in ${\mathrm{AdS}}_5\times S^5$, which is the giant magnon solution. We present two independent ways to obtain this correction: the first approach makes use of the algebraic curve description of the giant magnon. The second relies on the purely field-theoretical Lüscher formulas, which depend on the world-sheet$S$-matrix. We demonstrate the agreement to all orders in $(\Delta /\sqrt{\lambda }{)}^{-1}$ of these approaches, which, in particular, presents a further test of the $S$-matrix. We comment on generalizations of this method of computation to other string configurations.

Figure 1

The two leading processes contributing to the corrections of the dispersion relation in finite volume: The $F$-term (left-hand side) describes the contribution of a virtual particle loop. We shall see that it accounts for the leading contribution or $\mathcal{O}(e^{-2\pi \Delta /\sqrt{\lambda }})$ correction to the giant magnon one-loop energy. The $\mu$-term (right-hand side) is the effect of a particle splitting into two on-shell particles and computes the classical finite-size effects. It also contributes to the one-loop shift, but since it is $\mathcal{O}(e^{-2\pi \Delta /\sqrt{\lambda }\sin (p/2)})$, this contribution is subleading.

Figure 2

The small charge dyonic giant magnon in finite volume [21]: the endpoints of the log-cut between $X^{+}$ and $X^{-}$ develop small square root tails, separated by $\mathcal{O}(e^{-(\pi \mathcal{J}/(\sin p/2))})$, which will induce finite-size corrections of this order. For the classical finite-size corrections, these were the only contributions. However, as we explain in the main text, for the leading quantum correction there are more important corrections, which are of order $\mathcal{O}(e^{-2\pi \mathcal{J}})$.