We write an integral equation that incorporates finite corrections to the large spin asymptotics of N ¼ 4 supersymmetric Yang-Mills theory twist operators from the nonlinear integral equation. We show that these corrections are an all-loop result, not affected by wrapping effects, and agree, after determining the strong coupling expansion, with string theory predictions.

We compute the anomalous dimensions of field strength operators Tr F L in N = 4 SYM from an asymptotic nested Bethe ansatz to all-loop order. Starting from the exact solution of the one-loop problem at arbitrary L, we derive a single effective integral equation for the thermodynamic L → ∞ limit of these dimensions. We also include the recently proposed phase factor for the S-matrix of the planar AdS/CFT system. The terms in the effective equation corresponding to, respectively, the nesting and the dressing are structurally very similar. This hints at the physical origin of the dressing phase, which we conjecture to arise from the hidden presence of infinitely many auxiliary Bethe roots describing a nontrivial "filled" structure of the theory's BPS vacuum. We finally show that the mechanism for creating effective nesting/dressing kernels is quite generic by also deriving the integral equation for the all-loop dimension of a certain one-loop so(6) singlet state.There is much evidence that planar N = 4 SYM theory is integrable and that its spectral problem is therefore exactly solvable. It was shown by Minahan and Zarembo that the dilatation operator in the scalar matter sector at one loop can be mapped to the Hamiltonian of an integrable so(6) spin chain and hence its eigenvalues can be found with a Bethe ansatz [1]. This extends to the full set of operators, leading to an integrable non-compact nearest-neighbor super magnet [2]. The special "solvable" properties of the N = 4 model under dilatation were already hinted at by Lipatov in [3], and extend a rather generic if incomplete phenomenon in more general gauge theories such as QCD, as first shown by Belitsky, Braun, Derkachov, Korchemsky and Manashov [4].The concept of factorized scattering, one of the hallmarks of integrability, can be extended to higher loop orders [5], and to strong coupling [6], where the gauge theory is expected to be more suitably described by a superstring theory in a curved AdS 5 × S 5 background. Various analyses of this topic [7] led to a set of asymptotic all-loop Bethe equations for the full theory [8]. It should be stressed that quantum integrability remains to be proven in both gauge and string theory. In particular, on the gauge side one phenomenologically finds an integrable long-range spin chain, and the all-loop factorization of the multi-body magnon S-matrix into two-body processes currently has to be assumed. Correspondingly, symmetry fixes the magnon S-matrix only up to an overall phase factor [5,8,9]. The latter encodes our lack of understanding of the underlying microscopic integrable structure 1 . However, as was argued by Janik, the dressing phase may be constrained by invoking crossing-invariance 2 [11]. And indeed the string S-matrix satisfies crossing to the known [6, 12] orders [13]. A proposal for the complete structure of the dressing phase has recently been made in [14] by combining Bethe ansatz techniques for the all-order perturbative large spin limit of Wilson twist operators [15] with conjectures by Beisert, H...

We present a formula for the five-loop anomalous dimension of N = 4 SYM twist-three operators in the sl(2) sector. We obtain its asymptotic part from the Bethe Ansatz and finite volume corrections from the generalized Lüscher formalism, considering scattering processes of spin chain magnons with virtual particles that travel along the cylinder. The complete result respects the expected large spin scaling properties and passes non-trivial tests including reciprocity constraints. We analyze the pole structure and find agreement with a conjectured resummation formula. In analogy with the twist-two anomalous dimension at four-loops wrapping effects are of order (log 2 M/M 2 ) for large values of the spin.

We identify the gauge theory dual of a spinning string of minimal energy with spins S 1 , S 2 on AdS 5 and charge J on S 5 . For this purpose we focus on a certain set of local operators with two different types of covariant derivatives acting on complex scalar fields. We analyse the corresponding nested Bethe equations for the ground states in the limit of large spins. The auxiliary Bethe roots form certain string configurations in the complex plane, which enable us to derive integral equations for the leading and subleading contribution to the anomalous dimension. The results can be expressed through the observables of the sl(2) sub-sector, i.e. the cusp anomaly f (g) and the virtual scaling function B L (g), rendering the strong-coupling analysis straightforward. Furthermore, we also study a particular sub-class of these operators specialising to a scaling limit with finite values of the second spin at weak and strong coupling.