Nonplanar Integrability: Beyond the SU(2) Sector

Badr Awad Elseid Mohammed; Robert de Mello Koch; Stephanie Smith

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Nonplanar Integrability: Beyond the SU(2) Sector

Robert de Mello Koch, Badr Awad Elseid Mohammed, Stephanie Smith

High Energy Physics - Theory · Submitted 13 Jun 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

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arXiv ID1106.2483 (version 1)

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CitationBadr Awad Elseid Mohammed, Robert de Mello Koch, Stephanie Smith. "Nonplanar Integrability: Beyond the SU(2) Sector." arXiv:1106.2483 (2011). https://pith.science/paper/K3QUVCJG

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What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.94 · mentions code or data availability 0.01.

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We compute the one loop anomalous dimensions of restricted Schur polynomials with a classical dimension \Delta\sim O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We compute the one loop anomalous dimensions of restricted Schur polynomials with a classical dimension \Delta\sim O(N).	contribution 0.64	0.96	0.04	asserted 1.00
s1	The operators that we consider are labeled by Young diagrams with two long columns or two long rows.	contribution 0.86	0.52	0.02	asserted 1.00
s2	Simple analytic expressions for the action of the dilatation operator are found.	result 0.93	0.94	0.04	asserted 1.00
s3	The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory.	method 0.95	0.83	0.04	asserted 1.00
s4	The diagonalization of the dilatation operator reduces to solving five term recursion relations.	method 0.55	0.88	0.06	asserted 1.00
s5	The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials.	result 0.73	0.89	0.09	asserted 1.00
s6	This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.	interpretation 0.65	0.88	0.88	asserted 1.00

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Original abstract

We compute the one loop anomalous dimensions of restricted Schur polynomials with a classical dimension \Delta\sim O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

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References

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Handling Handles: Nonplanar Integrability in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory
hep-th 2017-11 conditional novelty 7.0 of 10

An integrability-based recipe reproduces the first non-planar correction to a four-point correlator in N=4 super Yang-Mills at one and two loops.
A double coset ansatz for integrability in AdS/CFT
hep-th 2012-04 accept novelty 6.0 of 10

The authors prove that a double coset of permutation groups organizes both the counting of string-brane configurations and the diagonalization of the one-loop dilatation operator for strings attached to giant gravitons.
Giant Graviton Oscillators
hep-th 2011-08 conditional novelty 6.0 of 10

Using a new U(p) Schur-Weyl duality construction, the authors compute the one-loop dilatation operator on restricted Schur polynomials with p long rows or columns and find it matches decoupled harmonic oscillators in ...

How the literature uses it

18 arXiv papers cite this in 32 sentences

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Papers citing it · 22 sentences without a confident label

Builds on it 2 papers

This was then extended beyond the su$(2)$ sector in [ref] and to an arbitrary number of rows in [ref] .

Nonplanar integrability at two loops · Introduction and Questions · 0.86
In [ref] , these results are extended beyond the $SU(2)$ sector for systems of two giant gravitons.

On subgroup adapted bases for representations of the symmetric group · Introduction · 0.84

Compares with it 2 papers

The comparison for this case is even more robust, since the BPS states in [ref] were found explicitly, by looking for eigenstates of one-loop dilatation operator, so we know precisely the dual BPS operators, that we are matching the excitations with.

Quantum states to brane geometries via fuzzy moduli spaces of giant gravitons · Half-BPS backgrounds · 0.97
This normalization for $D_2$ is a factor of 2 larger than the normalization used in [ref] .]

Nonplanar integrability at two loops · Spectrum · 0.90

Cites it as background 5 papers

The construction of the restricted Schur polynomial operators in various sectors, such as $su(2)$, $su(2|3),$ $sl(2)$ and $su(3)$ sectors, and their anomalous dimensions have been considered in the recent work for example [ref] .

Relation between large dimension operators and oscillator algebra of Young diagrams · Bases of operators and mixing of operators · 0.98
In the context of restricted Schur basis, there has been extensive work on the mixing due to the action of one-loop dilatation operator in the context of integrability of strings attached to giant gravitons [ref] .

Quantum states to brane geometries via fuzzy moduli spaces of giant gravitons · Discussion: labelling and construction of operators · 0.93
Generalizations of restricted Schur polynomials to fermion fields, gauge fields and three complex scalar fields have been studied in [ref] .

Counting paths with Schur transitions · Restricted Schur polynomials · 0.90

What it cites · 52 sentences without a confident label

Builds on 3 papers

The main goal of this article is to extend the results of [ref] beyond the $SU(2)$ sector.

Emergent Threebrane Lattices · Introduction · 1.00
The main goal of this article is to extend the results of [ref] beyond the $SU(2)$ sector.

Surprisingly Simple Spectra · Introduction · 1.00
The main goal of this article is to extend the results of [ref] beyond the $SU(2)$ sector.

Nonplanar Integrability · Introduction · 1.00

Uses the method or tool of 3 papers

To solve these recurrence relations, introduce the Hahn polynomial [ref]

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue · Diagonalization of the Dilatation Operator · 0.93
This factor arises after using the reduction rule of [ref] .

Giant Graviton Correlators from Dual SU(N) super Yang-Mills Theory · Action of the Dilatation Operator · 0.98
This factor arises after using the reduction rule of [ref] .

Giant Gravitons - with Strings Attached (I) · Action of the Dilatation Operator · 0.98

Compares with 2 papers

These are identical to the three term recursion relations that appear in [ref] .

Nonplanar Integrability · Diagonalization of the Dilatation Operator · 0.96
This is strongly reminiscent of the 1/2 BPS case where the Schur polynomials provide wave functions for fermions in a harmonic oscillator potential and further, these wave functions very naturally reproduce features of the geometries and the phase space [ref] .

Bubbling AdS space and 1/2 BPS geometries · Discussion · 0.89

Cites as background 8 papers

The intertwiner operators relevant for our study have been discussed in detail in [ref] .

Surprisingly Simple Spectra · Action of the Dilatation Operator · 0.85
In [ref] the BMN operators [ref] in an LLM background [ref] were considered.

Hints of Integrability Beyond the Planar Limit: Nontrivial Backgrounds · Introduction · 0.90
In [ref] the BMN operators [ref] in an LLM background [ref] were considered.

Strings in flat space and pp waves from ${\cal N}=4$ Super Yang Mills · Introduction · 0.90

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