@MISC{Ramadanovic08finitesize, author = {Bojan Ramadanovic and Gordon W. Semenoff}, title = {Finite size giant magnon}, year = {2008} }

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Abstract

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 5. 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 1 computations of short operators [16],[17] suggest that the most advanced form of the integrability Ansätz, due to Beisert, Eden and Staudacher [14], is likely valid only in the infinite size limit and it is spoiled by finite size effects. In the gauge theory, these effects are thought to stem from wrapping interactions [18],[19],[20]. A place where some progress has been made in studying finite size effects is the spectrum of a single magnon. The Bethe Ansätz implies that the energy spectrum of a single magnon (at least in infinite volume) has the form