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 AdS5×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.