Fine structure of the string spectrum in AdS 5 × S 5

Abstract

The spectrum of an infinite spinning string in AdS5 does not precisely match the spectrum of dual gauge theory operators, interpolated to the strong coupling regime with the help of Bethe-ansatz equations. We show that the mismatch is due to interactions in the string σ-model which cannot be neglected even at asymptotically large 't Hooft coupling. PACS numbers: Valid PACS appear hereAccording to the AdS/CFT correspondence [1][2][3], N = 4 supersymmetric Yang-Mills theory (SYM) and string theory in AdS 5 × S 5 have a common spectrum that continuously interpolates between the loop-corrected dimensional analysis at weak coupling and the string oscillator spectrum at strong coupling. The complete integrability of the AdS/CFT system makes the non-perturbative interpolation amenable to an exact description by methods of Bethe ansatz [4]. The string interpretation of the spectrum, however, is quite subtle and our goal is to find a potential resolution of these subtleties.We shall concentrate on a specific set of states related to twist-two operators tr ZD S + Z. Here Z is a complex scalar field in SYM and D + is the covariant derivative in light-cone direction. The twist operators constitute presumably the most studied sector of the SYM spectrum [5]. Their anomalous dimensions scale logarithmically with spin: ∆ S (λ) − S 2Γ cusp (λ) ln S, where the cusp anomalous dimension Γ cusp (λ) is a non-trivial function of the 't Hooft coupling λ = g 2 YM N , which can be computed non-perturbatively with the help of the Betheansatz equations [6]. On the string side, twist operators are described by a string spinning in the Anti-de-Sitter space [7]. When the spin is very large, the string becomes essentially infinite, extending all the way to the boundary. The energy density of this long string is equal to the cusp anomalous dimensions Γ cusp (λ).We will be interested in the spectrum of small fluctuations on top of the long string, which are dual to operators with extra field insertions, schematically:where Ψ i can be a field strength, a fermion or a scalar. Each insertion corresponds to an elementary excitation above the ground state. The spectrum of elementary excitations can be found exactly [8,9] by solving the Bethe-ansatz equations [6], and should agree at strong coupling with the spectrum of the string in light-cone gauge. A detailed comparison reveals, however, several mismatches [10]. Since the above operators have many uses, for instance they govern the collinear limits of scattering amplitudes [11], it is important to understand how these discrepancies are resolved.The string oscillation modes in light-cone gauge are two-dimensional massive particles, whose interactions are The fermions form four 2d Dirac spinors with eight degrees of freedom on shell. The SYM spectrum, continued to strong coupling, consists of [8]:(Field strength) (2) :Fermions (8) :where in brackets we indicated the number of degrees of freedom of each excitation. The agreement holds literally only for fermions. The heavy boson from AdS 3 is absent i...

Quantum dispersion relations for the Ad S 4 × ℂ ℙ 3 $$ Ad{S}_4\times \mathrm{\mathbb{C}}{\mathrm{\mathbb{P}}}^3 $$ GKP string

“…On the other hand, an identification of the massless modes and their dispersion relations turns out to be rather problematic in a perturbative approach. A similar mismatch has been already highlighted in [47] in the context of AdS 5 × S 5 . In that case the five massless scalars of the superstring sigma model are not in direct correspondence with the six massless degrees of freedom of the integrability description (and the low energy O(6) sigma model).…”

“…Eventually the spectrum consists of 4+4 excitations (spinons) acquiring a nonperturbative mass, which are identified with the holes of the integrability description. Furthermore, as was pointed out in [47] for the AdS 5 ×S 5 case, the heaviest scalar mode of the sigma model is not present in the integrability analysis as an elementary excitation. A similar issue is encountered in the AdS 4 × CP 3 setting and in the following we comment on the interpretation of such a mode.…”

“…mode, the binding energy of gauge excitations bound states and the corrections to the dispersion relation of massless states was presented in [47].…”

Strong Wilson polygons from the lodge of free and bound mesons

“…In fact, several authors (e.g. [22,27,49]) have supposed that this particle (allegedly formed by two mass 1 fermions) may exit the physical domain at finite values of the coupling constant, or, in other words, become virtual. In the context of the BSV series, it was shown in [33] that, in the infinite coupling limit, a mass 2 contribution arises from a pole in the state of a small fermion and a small antifermion (similar phenomena are also said to happen for more fermions).…”

“…In specific, we deliberately inverted the order of collinear limit (the series) with the large coupling one in an educated manner: in fact, motivated by a pole in the fermion-antifermion S-matrix at generic coupling, we have performed a bootstrap on the (asymptotic) Bethe Anstaz equations of [42] and thus shown the existence of a new mass 2 particle in two dimensions (cf. the discussion below and [27,33,[47][48][49]), a meson, along with its infinite bound states, only at strong coupling. Correspondingly we have modified the BSV series at large coupling by adding the contributions of these particles (whilst initially they were not in the spectrum).…”