Finite size giant magnon

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 Z M -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]- [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]- [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 [7]. Beginning with this limit, a strategy advocated by Staudacher [8], Beisert showed that a residual SU(2|2) 2 supersymmetry and integrability determine the N = 4 S-matrix up to a phase [9], [10]. More recent work constrains [11] and essentially computes this phase [12]-[17].

Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries

“…This was used in [63] to settle an issue of gauge noninvariance (dependence of the magnon energy on the light-cone gauge fixing parameter, once finite-size effects are considered) which had previously arisen in the AdS 5 × S 5 case [64]. It was later argued that single magnons in N = 4 SYM can always be thought of as living on the orbifolded theory [65]. Recently, TBA equations and wrapping effects (up to next-to-leading order) were considered for a particular orbifold theory in [34].…”

Giant magnons inAdS4×CP3: embeddings, charges and a Hamiltonian

“…The apparent gauge-dependence of the results of [29] was resolved by [31], using the fact that the solutions are periodic both on the worldsheet and in the azimuthal angle on the sphere, and so can be viewed as wound strings on S 2 /Z n . [31,32] The scattering of finite-J magnons was studied by [33], using the connection to sine-gordon theory in finite volume. The finite-J generalisations of the basic giant magnon are still solutions moving on S 2 , and so one can place them into CP 3 using either of the maps presented in sections 5.1 and 5.2 above.…”

Anomalous dimension with wrapping at four loops in SYM

“…In [17] wrapping effects in some simple spin chain models 1 were studied. On the string theory side, finite size contributions have been analyzed in several recent papers [18][19][20][21][22][23][24][25].…”