Wrapping interactions at strong coupling: The giant magnon

Abstract

We derive generalized Lüscher formulas for finite size corrections in a theory with a general dispersion relation. For the AdS 5 × S 5 superstring these formulas encode leading wrapping interaction effects. We apply the generalized µ-term formula to calculate finite size corrections to the dispersion relation of the giant magnon at strong coupling. The result exactly agrees with the classical string computation of Arutyunov, Frolov and Zamaklar. The agreement involved a Borel resummation of all even loop-orders of the BES/BHL dressing factor thus providing a strong consistency check for the choice of the dressing factor. *

Resurgence of the dressing phase for AdS5 × S5

“…Note that this analysis is very similar to the one carried out in [21], particularly the discussion in their appendix C. However, in [21] the authors considered only the magnon regime for which x 1 and x 2 take a rather special form at strong coupling, thus allowing to simplify equation (8.7). Furthermore, since the authors were interested in extracting the leading contribution they focused only on the case where z is an even integer.…”

“…In this way they found a connection between the strong and weak coupling coefficients of the dressing phase, reminiscent of the analytic continuation conjectured in [14]. Similarly the leading contribution to the magnons dressing phase in the strong coupling regime was obtained in [21] via Borel resummation of a particular class of terms in the asymptotic limit. Finally in [22] the authors expanded the dressing phase of [14] reproducing precisely the asymptotic strong coupling coefficients.…”

“…To obtain the JHEP01(2017)055 strong coupling regime, these authors did not use the contour integral type of argument utilized in [14], but rather implemented a suitable ad hoc regularization procedure to expand the integrand of the Beisert-Eden-Staudacher dressing phase and obtained back the formal asymptotic expansion studied in [13]. Although the results of [20,21] and [22] suggest that the dressing phase proposed in [14] (that we will call BES in what follows) has the correct properties to interpolate between the weak and strong coupling regime, to our mind no rigorous and complete treatment of the resummation procedure of the full strong coupling asymptotic expansion of σ exists so far.…”

A next-to-leading Lüscher formula

“…We found only a posteriori justifications to this fact: the exact Bethe equations would reduce to the asymptotic ones if the probe particle belongs to the rotated theory, or, in other words, particles of physical and rotated theories seem to interact only through the S-matrix, as it is confirmed by the results of this paper. 3 Even though in the relativistic case the space-time rotated theory is the same as the physical one, in view of the generalization to the AdS/CF T case, we already start to use here the terminology related to the non-relativistic case, were the rotated theory is different from the physical one and is called mirror theory, first introduced in [10]. 4 One should be careful with the analytical continuation to the mirror theory, since shifting back the integration contour to the real line, poles of the S-matrix, corresponding to bound-states between mirror and physical particles, could be met, giving additional contributions to the energy, called µ-terms.…”

“…In particular, the study of the finite-volume corrections [2][3][4][5][6][7][8][9] for the anomalous dimensions/string energies spectrum of AdS 5 /CF T 4 , has culminated in the formulation of the so-called Thermodynamic Bethe Ansatz (TBA) equations and Y-system [10][11][12][13][14][15][16][17][18], which in principle govern the spectrum exactly at any order of the coupling constant and the volume parameter. Very recently, the TBA equations have been reduced first to few non-linear integral (so-called FiNLIE) equations [19] (see [17,18,[20][21][22][23][24][25][26][27] for some previous developments in that direction), then to an impressively simple set of Riemann-Hilbert equations [28].…”