The Dilatation Operator of Conformal N=4 Super Yang-Mills Theory

Abstract

We argue that existing methods for the perturbative computation of anomalous dimensions and the disentanglement of mixing in N = 4 gauge theory can be considerably simplified, systematized and extended by focusing on the theory's dilatation operator. The efficiency of the method is first illustrated at the one-loop level for general non-derivative scalar states. We then go on to derive, for pure scalar states, the two-loop structure of the dilatation operator. This allows us to obtain a host of new results. Among these are an infinite number of previously unknown two-loop anomalous dimensions, new subtleties concerning 't Hooft's large N expansion due to mixing effects of degenerate single and multiple trace states, two-loop tests of various protected operators, as well as two-loop non-planar results for two-impurity operators in BMN gauge theory. We also put to use the recently discovered integrable spin chain description of the planar one-loop dilatation operator and show that the associated Yang-Baxter equation explains the existence of a hitherto unknown planar ``axial'' symmetry between infinitely many gauge theory states. We present evidence that this integrability can be extended to all loops, with intriguing consequences for gauge theory, and that it leads to a novel integrable deformation of the XXX Heisenberg spin chain. Assuming that the integrability structure extends to more than two loops, we determine the planar three-loop contribution to the dilatation operator.

54 pages, 5 figures, v2: references added, small textual changes, v3: to appear in Nucl. Phys. B, v4: zeros in (D.21), signs in (F.4) corrected

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• The Bethe-Ansatz for N=4 Super Yang-Mills
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• Exact anomalous dimensions of {\cal N}=4 Yang-Mills operators with large R charge
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• Operator Mixing and the BMN Correspondence
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• More comments on superstring interactions in the pp-wave background
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• A Calculation of the plane wave string Hamiltonian from N=4 super-Yang-Mills theory
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• Explicit Formulas for Neumann Coefficients in the Plane-Wave Geometry
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• Bound States in the Mirror TBA
• Wrapping Interactions and the Konishi Operator
• Spiky Strings and Giant Holes
• N=4 Super-Yang-Mills Theory, QCD and Collider Physics
• Integrability and MHV diagrams in N=4 supersymmetric Yang-Mills theory
• Chaotic spin chains in AdS/CFT
• Non-planar anomalous dimension of twist-2 operators: higher moments at four loops
• Superstring in the plane-wave background with RR flux as a conformal field theory
• Finite-size corrections to the rotating string and the winding state
• Issues on magnon reflection
• More Pendants for Polya: Two loops in the SU(2) sector
• New aspects of the BMN correspondence beyond the planar limit
• Emergent Classical Strings from Matrix Model
• Quarter-BPS states, multi-symmetric functions and set partitions
• Coherent state operators, giant gravitons, and gauge-gravity correspondence
• Nesting and Dressing
• Twist-2 at five loops: Wrapping corrections without wrapping computations
• The two-loop dilatation operator of N=4 super Yang-Mills theory in the SO(6) sector
• Hints of Integrability Beyond the Planar Limit: Nontrivial Backgrounds
• On string integrability. A journey through the two-dimensional hidden symmetries in the AdS/CFT dualities
• Recursion Relations for Long-Range Integrable Spin Chains with Open Boundary Conditions
• DLCQ String Spectrum from $\mathcal{N}=2$ SYM Theory
• On Three-point Functions in the AdS_4/CFT_3 Correspondence
• The exactly solvable spin Sutherland model of B_N type and its related spin chain
• Nonlocal charges of T-dual strings
• Double-logs, Gribov-Lipatov reciprocity and wrapping
• Decay widths of three-impurity states in the BMN correspondence
• Fermions and scalars in $\mathcal{N}=4$ Wilson loops at strong coupling and beyond
• Exact solutions for N-magnon scattering
• On twist-two operators in N=4 SYM
• Introduction to Integrability and One-point Functions in $\mathcal{N}=4$ SYM and its Defect Cousin
• Anomalous Dimensions of Heavy Operators from Magnon Energies
• CFT4 as SO(4,2)-invariant TFT2
• Coherent state excitations and string-added coherent states in gauge-gravity correspondence
• Relation between large dimension operators and oscillator algebra of Young diagrams
• Correlators on the wall and ${\mathfrak{sl}}_n$ spin chain
• Magnons and BFKL
• Two-dimensional Yang-Mills theory, Painleve equations and the six-vertex model
• Sum rules for higher twist sl(2)operators in N=4 SYM
• Spin bits at two loops
• The Leading Order Dressing Phase in ABJM Theory
• Yangians in Integrable Field Theories, Spin Chains and Gauge-String Dualities
• Three-point functions of BMN operators at weak and strong coupling II. One loop matching
• New Penrose Limits and AdS/CFT
• Note on Integrability of Marginally Deformed ABJ(M) Theories
• Integrable spin chains and cellular automata with medium range interaction
• Wrapping corrections for long range spin chains
• Wilson Line Correlators in N=4 Non-commutative Gauge Theory on S^2 x S^2
• Spin chain for the deformed ABJM theory
• Zero Modes for the Boundary Giant Magnons
• Testing the nested light-cone Bethe equations of the AdS_5 x S^5 superstring
• From nesting to dressing
• Review of AdS/CFT Integrability, Chapter I.2: The spectrum from perturbative gauge theory
• Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM
• SU(2|2) for Theories with Sixteen Supercharges at Weak and Strong Coupling
• The Complete One-loop Spin Chain of N =1 SQCD
• The contribution of scalars to $\mathcal{N}=4$ SYM amplitudes II: Young tableaux, asymptotic factorisation and strong coupling
• Negative anomalous dimensions in N=4 SYM
• Subleading corrections to the Double Coset Ansatz preserve integrability
• Nonplanar Integrability and Parity in ABJ Theory
• Gauge Theories as String Theories: the First Results
• On the Integrability of String Theory in AdS_5 x S^5
• Strong coupling anomalous dimensions of $\mathcal{N}=4$ super Yang-Mills
• One-Loop Non-Planar Anomalous Dimensions in Super Yang-Mills Theory
• Strong Wilson polygons from the lodge of free and bound mesons
• Integrability and the AdS/CFT correspondence
• Constructing Non-Relativistic AdS${}_5$/CFT${}_4$ Holography
• Non-perturbative effects in the BMN limit of N=4 supersymmetric Yang-Mills
• Strong coupling from the Hubbard model
• Quasiclassical Geometry and Integrability of AdS/CFT Correspondence
• Asymptotic Bethe equations for open boundaries in planar AdS/CFT
• Chiral primaries in the Leigh-Strassler deformed N=4 SYM -- a perturbative study
• Higher Charges in Dynamical Spin Chains for SYM Theory
• On the breakdown of perturbative integrability in large N matrix models
• Analytic Bethe Ansatz and Baxter equations for long-range psl(2|2) spin chain
• The dual string sigma-model of the SU_q(3) sector
• Is there a tower of charges to be discovered?
• Statistical mechanics for dilatations in N=4 super Yang--Mills theory
• N=1 SQCD and the Transverse Field Ising Model
• Deviation from Alday-Maldacena Duality For Wavy Circle
• Graph duality as an instrument of Gauge-String correspondence
• Giant Holes
• N=4 SYM Quantum Spectral Curve in BFKL regime
• Phases of Thermal N=2 Quiver Gauge Theories
• On dilatation operator for a renormalizable theory
• On Yangian symmetry of scattering amplitudes and the dilatation operator in N=4 super Yang-Mills
• Spinning Strings on Deformed AdS_5 x T^{1,1} with NS B-field
• QCD-like properties for anomalous dimensions in N=4 SYM
• Lifting integrable models and long-range interactions
• Partition functions of Polychronakos like spin chains associated with polarized spin reversal operators
• S-duality invariant perturbation theory improved by holography
• Generating Functions for Giant Graviton Bound States
• Four-loop anomalous dimensions in Leigh-Strassler deformations
• Interactions as conformal intertwiners in 4D QFT
• Aspects of Integrability in N =4 SYM
• Excited States of Open Strings From $\mathcal{N}=4$ SYM
• Oscillating Multiple Giants
• Giant Spinons
• Four-Impurity Operators and String Field Theory Vertex in the BMN Correspondence
• The Ising model and planar N=4 Yang-Mills
• Non-planar operator mixing by Brauer representations
• Wrapping effects in supersymmetric gauge theories
• Integrable systems and AdS/CFT duality
• Quantum Strings and Bethe Equations
• Interacting Double Coset Magnons
• Nonplanar Four-Loop Anomalous Dimensions of Twist-Two Operators in N=4 Super Yang-Mills Theory: Higher Moment, General Result, and Cusp Anomalous Dimension
• Appearance of branched motifs in the spectra of $B{C}_N$ type Polychronakos spin chains
• Rationality of the Anomalous Dimensions in N=4 SYM theory
• DGLAP and BFKL equations in $\mathcal{N}=4$ SYM: from weak to strong coupling
• Supersymmetric analogue of BC_N type rational integrable models with polarized spin reversal operators
• BMN operators with a scalar fermion pair and operator mixing in N=4 Super Yang-Mills Theory
• Analytic Bethe Ansatz Solutions for Highest States in the su(1|1) and su(2) Sectors
• Super Rogers-Szegö polynomials associated with $B{C}_N$ type of Polychronakos spin chains
• Perturbative 4D conformal field theories and representation theory of diagram algebras
• Supergraphs and the cubic Leigh-Strassler model
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Operator mixing in N = 4 SYM: The Konishi anomaly re-re-visited
• The SU(2) Long Range Bethe Ansatz and Continuous Integrable Systems
• Gauge-String Dualities and Some Applications
• Dilatation Operator and Space-time Geometry
• Fermion zero modes for the mixed-flux AdS${}_3$ giant magnon
• Exact Results in Supersymmetric Gauge Theories
• Lee-Yang Model in Presence of Defects
• Commuting Conformal and Dual Conformal Symmetries in the Regge limit
• The Hagedorn temperature in a decoupled sector of AdS/CFT
• Large N anomalous dimensions for large operators in Leigh-Strassler deformed SYM
• Multi-trace Correlators from Permutations as Moduli Space
• Semiclassical quantization of the mixed-flux AdS${}_3$ giant magnon
• On emergence in gauge theories at the 't Hooft limit
• Symmetry structure of the interactions in near-BPS corners of $\mathcal{N}=4$ super-Yang-Mills
• Conformal Mapping of Non-Lorentzian Geometries in SU(1,2) Conformal Field Theory
• A hitchhiker's guide to quantum field theoretic aspects of $\mathcal{N}=4$ SYM theory and its deformations
• Fermionic contribution to the anomalous dimension of twist-2 operators in N=4 SYM theory, critical indices and integrability
• Splitting of Folded Strings in AdS_4*CP^3
• Supersymmetric String Configurations in Ad${\mathbb{S}}_5\times {\mathbb{S}}^5$
• A Young diagram expansion of the hexagonal Wilson loop (amplitude) in $\mathcal{N}=4$ SYM
• Matrix at slow roll: Nonrelativistic and Perturbative
• Gravitational realization of magnons in a ferromagnetic spin chain
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• On interpolating anomalous dimension of twist-two operators with general spins