We derive the perturbative five loop anomalous dimension of the Konishi operator in N = 4 SYM theory from the integrable string sigma model by evaluating finite size effects using Luscher formulas adapted to multimagnon states at weak coupling. In addition, we derive the five loop wrapping contribution for the L=2 single impurity state in the beta deformed theory, which may be within reach of a direct perturbative computation. The Konishi expression exhibits two new features - a modification of Asymptotic Bethe Ansatz quantization and sensitiveness to an infinite set of coefficients of the BES/BHL dressing phase. The result satisfies nontrivial self-consistency conditions - simple transcendentality structure and cancellation of mu-term poles. It may be a testing ground for the proposed AdS/CFT TBA systems.Comment: 45 pages, 1 figure; v2: refs added; v3: published versio

Using the recently proposed set of discontinuity relations we translate the AdS/CFT Y-system to TBA integral equations and quantization conditions for a large subset of excited states from the sl(2) sector of the AdS 5 ×S 5 string σ-model. Our derivation provides an analytic proof of the fact that the exact Bethe equations reduce to the BeisertStaudacher equations in the asymptotic limit. We also construct the corresponding Tsystem and show that in the language of T-functions the energy formula reduces to a single term which depends on a single T-function.

Using the linearized TBA equations recently obtained in [arXiv:1002.1711] we show analytically that the 5-loop anomalous dimension of the Konishi operator agrees with the result obtained previously from the generalized Luscher formulae. The proof is based on the relation between this linear system and the XXX model TBA equations.Comment: appendix added (published version