Finite size giant magnon

The quantization of the giant magnon away from the infinite size limit is discussed. We argue that this quantization inevitably leads to string theory on a ZM -orbifold of S . This is shown explicitly and examined in detail in the near plane-wave limit. A significant amount of work on the AdS/CFT correspondence [1],[2],[3] has been inspired the idea that the planar limit of N = 4 Yang-Mills theory and its string dual might be integrable models which would be completely solvable using a Bethe Ansätz [4],[5],[6]. Computation of the conformal dimensions of composite operators in N = 4 Yang-Mills theory can be mapped onto the problem of solving an SU(2, 2|4) spin chain. It is known that the spin chain simplifies considerably in the limit of infinite length where dynamics are encoded in the scattering of magnons and integrability would imply a factorized S-matrix. Beginning with this limit, a strategy advocated by Staudacher [7], Beisert showed that a residual SU(2|2)2 supersymmetry and integrability determine the N = 4 S-matrix up to a phase [8],[9]. More recent work constrains [10] and essentially computes this phase [11]-[15]. An important problem that the integrability program would eventually have to address is that of finite size corrections. In fact, recent four-loop