The complete one-loop dilatation operator of super-Yang–Mills theory

Abstract

We continue the analysis of hep-th/0303060 in the one-loop sector and present the complete psu(2, 2|4) dilatation operator of N = 4 Super YangMills theory. This operator generates the matrix of one-loop anomalous dimensions for all local operators in the theory. Using an oscillator representation we show how to apply the dilatation generator to a generic state. By way of example, we determine the planar anomalous dimensions of all operators up to and including dimension 5.5, where we also find some evidence for integrability. Finally, we investigate a number of subsectors of N = 4 SYM in which the dilatation operator simplifies. Among these we find the previously considered so(6) and su(2) subsectors, a su(2|4) subsector isomorphic to the BMN matrix model at one-loop, a su(2|3) supersymmetric subsector of nearly eighth-BPS states and, last but not least, a non-compact sl(2) subsector whose dilatation operator lifts uniquely to the full theory.

Higher impurity AdS/CFT correspondence in the near-BMN limit

“…In AdS 5 × S 5 , the Lagrangian takes the form 12) where Γ µ are SO(9, 1) Dirac gamma matrices, η µν is the SO(9, 1) Minkowski metric and s IJ = diag(1, −1). The worldsheet fermi fields θ I (I, J = 1, 2) are two SO(9, 1) MajoranaWeyl spinors of the same chirality Γ 11 θ I = θ I .…”

“…(4.9) and (4.20)). The one-loop spin chain Hamiltonian for this sector has been derived by Beisert [12] in a representation where there is a lattice site assigned to each Z field, and each site supports a harmonic oscillator whose level of excitation counts the number of D operators acting on that Z insertion. The raising operator a † i therefore corresponds to the insertion of a derivative at the i th lattice site:…”

Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings

“…The energy of these semiclassical strings admits an expansion in the effective coupling λ/J 2 , with λ the 't Hooft coupling of the gauge theory, suggesting the possibility of a precise comparison between string energies and anomalous dimensions of large N = 4 Yang-Mills operators. However operator mixing turned the computation of anomalous dimensions for large N = 4 operators into a formidable task, until the illuminating identification of the planar dilatation operator [5,6] with the Hamiltonian of an integrable quantum spin chain [7,8]. The spectrum of anomalous dimensions for large operators became then computable using the powerful technique of the Bethe ansatz, and complete agreement with the energies of semiclassical string states was found at the first two leading orders in λ [9]- [25].…”

Planar plane-wave matrix theory at the four loop order: Integrability without BMN scaling

“…In a nutshell it states that the dilatation operator of the planar gauge theory, which yields the anomalous dimensions of local composite operators in the conformal quantum field theory, is given by a long-range integrable spin chain Hamiltonian [3,4,5,6,7,8] 1 . This statement is by now firm at the one-loop level for the full theory in form of a (non-compact) su(2, 2|4) super spin-chain [5] and has been extended to the three-loop level in a closed supersymmetric su(2|3) subsector [6]. In this picture the loop order of the considered dilatation operator is linked to the spread of the local spin interactions of the spin-chain Hamiltonian.…”