Surprisingly Simple Spectra

Abstract

The large N limit of the anomalous dimensions of operators in $\mathcal{N}=4$ super Yang-Mills theory described by restricted Schur polynomials, are studied. We focus on operators labeled by Young diagrams that have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with $n\ne 2$ columns. The planar approximation does not capture the large N dynamics. For operators built with 2, 3 or 4 impurities the dilatation operator is explicitly evaluated. In all three cases, in a certain limit, the dilatation operator is a lattice version of a second derivative, with the lattice emerging from the Young diagram itself. The one loop dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of $8g_{YM}^2$ and there are interesting degeneracies in the spectrum. The spectrum we obtain for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. This equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian defined by the one loop large $N$ dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.

34 pages

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• Giant Graviton Oscillators
• A double coset ansatz for integrability in AdS/CFT
• Structure Constants in $\mathcal{N}=4$ SYM at Finite Coupling as Worldsheet $g$-Function
• From Large N Nonplanar Anomalous Dimensions to Open Spring Theory
• Yangians, S-matrices and AdS/CFT
• Restricted Schur Polynomials for Fermions and integrability in the su(2|3) sector
• Non-planar Anomalous Dimensions in the sl(2) Sector
• Correlation functions and representation bases in free N=4 Super Yang-Mills
• Operators, Correlators and Free Fermions for SO(N) and Sp(N)
• A basis for large operators in N=4 SYM with orthogonal gauge group
• BPS Chaos
• Nonplanar integrability at two loops
• The giant graviton on AdS_{4} x CP^{3} - another step towards the emergence of geometry
• LLM Magnons
• Orthogonal Bases of Invariants in Tensor Models
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• Exciting LLM Geometries
• Flavour singlets in gauge theory as Permutations
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• Relation between large dimension operators and oscillator algebra of Young diagrams
• Central Charges for the Double Coset
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• Young diagrams, Brauer algebras, and bubbling geometries
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• Restricted Schurs and correlators for SO(N) and Sp(N)
• Negative anomalous dimensions in N=4 SYM
• Nonplanar Integrability and Parity in ABJ Theory
• Orthogonal Schurs for Classical Gauge Groups
• Gauge Invariants and Correlators in Flavoured Quiver Gauge Theories
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• Novel charges in CFT's
• Generating Functions for Giant Graviton Bound States
• Correlation functions of determinant operators in conformal fishnet theory
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• Heavy Operators in Superconformal Chern-Simons Theory
• Interacting Emergent Yang-Mills theory
• Eigenvalue Dynamics for Multimatrix Models
• Large N anomalous dimensions for large operators in Leigh-Strassler deformed SYM
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