All loop BMN state energies from matrices

David Berenstein; Diego H. Correa; Samuel E. Vazquez

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All loop BMN state energies from matrices

David Berenstein, Diego H. Correa, Samuel E. Vazquez

High Energy Physics - Theory · Submitted 1 Sep 2005

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

The authors' main claim In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling.

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We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S^3. For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S^5 which we identify with the sphere on the dual geometry AdS_5x S^5. We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N=4 SYM planar spin chain model.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S^3.	contribution 0.91	0.85	0.03	asserted 0.97
s1	For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory.	background 0.92	0.06	0.03	hedged possibility 0.90
s2	In the large N limit the ground state corresponds to a distribution of eigenvalues on a S^5 which we identify with the sphere on the dual geometry AdS_5x S^5.	result 0.61	0.73	0.07	asserted 0.98
s3	We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively.	method 0.85	0.89	0.03	asserted 0.96
s4	To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom.	method 0.90	0.58	0.05	hedged possibility 0.64
s5	We then consider a state with two string bits and large angular momentum J on the sphere.	contribution 0.77	0.79	0.03	asserted 0.80
s6	In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling.	result 1.00	0.94	0.75	asserted 1.00
s7	We also find some new problems with the all loop Bethe Ansatz conjecture of the N=4 SYM planar spin chain model.	result 0.97	0.91	0.08	asserted 1.00

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Original abstract

We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S^3. For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S^5 which we identify with the sphere on the dual geometry AdS_5x S^5. We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N=4 SYM planar spin chain model.

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References

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Cited by 10 Pith papers

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

Giant Magnons
hep-th 2006-04 accept novelty 8.0 of 10

Magnons in the planar N=4 SYM spin chain are shown to correspond to giant strings on S5, with momentum equal to the angular separation of the string's endpoints, yielding the strong-coupling dispersion and S-matrix.
Reflecting magnons
hep-th 2007-08 accept novelty 7.0 of 10

Symmetry fixes the boundary reflection matrix of magnons off maximal giant gravitons up to a phase, verified at two loops in weak coupling and computed at strong coupling from classical string solutions.
Dyonic Giant Magnons
hep-th 2006-05 accept novelty 7.0 of 10

A two-parameter family of classical string solutions on AdS5 x S5 reproduces the exact BPS dispersion relation for magnon bound states from a purely classical calculation.
Submatrix deconfinement and small black holes in AdS
hep-th 2018-06 conditional novelty 6.0 of 10

Free gauged matrix quantum mechanics at intermediate energy exhibits partial deconfinement, with a U(M) sub-block whose size scales as the square root of the energy carrying the entropy, suggesting a smooth string-to-...
Giant gravitons: a collective coordinate approach
hep-th 2013-01 conditional novelty 6.0 of 10

Giant gravitons in AdS/CFT are described by a collective coordinate whose effective action is a first-order harmonic oscillator, and strings between separated giants acquire a mass proportional to the squared distance...

Show 5 more citing papers

Three-dimensional N=6 SCFT's and their membrane dynamics
hep-th 2008-08 conditional novelty 6.0 of 10

Quantizing the eigenvalue distribution of the ABJM theory on S^2 x R reproduces the dual M-theory geometry and yields the giant magnon dispersion relation sqrt((l+1/2)^2 + h(lambda) sin^2(p/2)), with the momentum depe...
Multi-matrix models and emergent geometry
hep-th 2008-05 conditional novelty 6.0 of 10

Two matrix-ratio observables distinguish commuting (geometric) from non-commuting (non-geometric) strong-coupling phases in solvable multi-matrix models, with emergent geometry requiring a mass term to regulate an inf...
A Test of the AdS/CFT Correspondence Using High-Spin Operators
hep-th 2006-11 conditional novelty 6.0 of 10

Numerically solving the BES integral equation reproduces the string-theory prediction f(g) = 4g - 3 ln2/pi at strong coupling to six digits and yields a new estimate for the 1/g coefficient.
Dressing the Giant Magnon
hep-th 2006-07 accept novelty 6.0 of 10

The dressing method yields explicit classical string solutions for scattering and bound states of one-, two-, and three-charge giant magnons on R x S^5, together with their dispersion relations.
Supersymmetric states of N=4 Yang-Mills from giant gravitons
hep-th 2006-06 accept novelty 6.0 of 10

Quantizing Mikhailov's 1/8 BPS giant graviton solution space yields the Hilbert space of N bosons in a 3d harmonic oscillator, matching the classical chiral ring of N=4 Yang-Mills and supporting the conjecture that th...

How the literature uses it

24 arXiv papers cite this in 67 sentences

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Papers citing it · 40 sentences without a confident label

Contests it 1 paper

Also, the convergence to large $N$ is slow, so the dependence on $q=Q/N$ is not reliable in the paper [ref] .

The shape of emergent quantum geometry from an N=4 SYM minisuperspace approximation · A scaling relation: size of the hole for varying $q$ at fixed $N$ · contest 0.76

Builds on it 2 papers

Here we are copying the same arguments presented in [ref] , with a modified notion of what $Z$ is.

Strings on conifolds from strong coupling dynamics, part I · The conifold and BMN limits · 0.81
For $Q>1$ , this formula is a generalisation of the exact [ref] magnon dispersion relation obtained in [ref] (see also [ref] ).

Aspects of Integrability in AdS/CFT Duality · The asymptotic states · 0.92

Uses its method or tool 6 papers

We will follow closely the techniques developed in [ref] .

Giant magnon bound states from strongly coupled N=4 SYM · Computation of the off-diagonal oscilator energies · 1.00
It is possible to study the off-diagonal excitations of this matrix model in a perturbative way, by treating them as free harmonic oscillators whose mass depends on the diagonal modes [ref] .

S-duality and the giant magnon dispersion relation · Introduction · 0.90
For this we follow the prescription of [ref] , and quantize the off-diagonal modes in the background of the diagonal eigenvalue distribution.

Black Hole Production from High Energy Scattering in AdS/CFT · The Strong Coupling Description · 0.96

Uses its data or values 3 papers

The effective Hamiltonian for this setup is a matrix quantum mechanics of six hermitian $N×N$ matrices $X⃗$, and their momenta [equation] where the coupling constant $A$ has been calculated in [ref] .

Numerical tests of AdS/CFT at strong coupling · Introduction · 0.92
The radius of the sphere was calculated in [ref] and it is given by $r_0 = √(N/2)$.

Black Hole Production from High Energy Scattering in AdS/CFT · The Strong Coupling Description · 0.99
The radius was computed in [ref] .

The shape of emergent quantum geometry from an N=4 SYM minisuperspace approximation · Wave functions and Geometric observables · 0.92

Compares with it 3 papers

For two impurities, and for special values of momenta, the perturbative spectrum of the two giant magnon bound state obtained in [ref] is recovered[A matrix derivation of this result has also been obtained [ref] in the context of an approach based on mutually commuting matrices [ref] .

Strongly coupled large N spectrum of two matrices coupled via a Yang-Mills interaction · More impurities and conclusion · 0.86
In the four-dimensional version of this story, this has a delta function support as explained in [ref] .

Three-dimensional N=6 SCFT's and their membrane dynamics · Eigenvalue gases · 0.80
It might be worthwhile to compare this with an approach [ref] , centering around the Berenstein's conjecture [ref] on the strongly coupled $N=4$ SYM on $ℝ×S^3$.

On the shape of a D-brane bound state and its topology change · Supersymmetric matrix models · 0.92

Cites it as support 3 papers

All off diagonal fluctuations are massive [ref] as are all the higher harmonics on $S^3$ (this second point follows from the analysis in [ref] ).

Strings on conifolds from strong coupling dynamics: quantitative results · The emergent geometry · 0.99
More evidence has shown that this approach may account systematically for the strong coupling expansion of $N=4$ SYM from first principles via purely field theoretical results [ref] (see also [ref] ).

Giant magnon bound states from strongly coupled N=4 SYM · Introduction · 1.00
One can reproduce exactly the energies of various strings in $_5×S^5$ this way [ref] .

Small AdS black holes from SYM · Semiclassical setup: setting the initial conditions · 0.96

Cites it as background 6 papers

To understand how the calculation done in the previous section relates to the spectrum of giant magnons, we need to review some of the results found in [ref] .

Giant magnon bound states from strongly coupled N=4 SYM · Chain localization argument and strong coupling · 0.96
When studying the action of the symmetries, it has proven to be useful to have in mind the physical picture for the extra central charges suggested by the classical string theory analysis in [ref] (see also [ref] for a related picture).

Reflecting magnons · Introduction · 0.84
These equations of motion have been considered in some detail in, for instance, [ref] .

Multi-matrix models and emergent geometry · `Supersymmetrised' models · 0.97

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