$N^3$-behavior from 5D Yang-Mills theory

A. Nedelin; J. A. Minahan; J. Kallen; M. Zabzine

REVIEW

$N^3$-behavior from 5D Yang-Mills theory

J. Kallen, J. A. Minahan, A. Nedelin, M. Zabzine

High Energy Physics - Theory · Submitted 16 Jul 2012 · 25 references

Pith's verdict: The N^3 coefficient is a real, checkable number and the saturation approximation is fine; the soft spot is that the model computed is not quite the maximally supersymmetric theory the title advertises.

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arXiv ID1207.3763 (version 4)

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CitationA. Nedelin, J. A. Minahan, J. Kallen, M. Zabzine. "$N^3$-behavior from 5D Yang-Mills theory." arXiv:1207.3763 (2012). https://pith.science/paper/4JVPU2KA

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Editor's note

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Short version: the N^3 scaling here is real and the arithmetic checks out, but the title's 'maximally supersymmetric' carries more weight than the model actually supports, and the authors know it. Send it out, but make sure a referee presses on the identification rather than on the saddle point, which is fine. On the saddle point: the reader's weakest-assumption worry does not land. The exact equation (3.3) differs from the linear form (3.5) by terms that are exponentially small once |phi_i - phi_j| is large, and the solution (3.6) spreads the eigenvalues over O(lambda), so the pairs that dominate the sum are saturated. The near-neighbour corrections are odd in x and cancel by symmetry, leaving a relative O(1/lambda) correction. The N^3 coefficient is not at risk. What is new and worth having: the strong-coupling evaluation of the localization model from [12] giving F = -27 g^2 N^3/(512 pi r), the Z_k quiver result N^3/k^2, and the fundamental-matter regime N^2 for M <= 2N with destabilization above. The mechanism is elementary, and the paper says so, but the coefficient is a concrete target. The soft spots, in order of weight. First, the abstract (s0) says '5D maximally supersymmetric Yang-Mills on S^5', while section 2 says the computed model is N=1 SYM with an adjoint hypermultiplet, 8 supercharges, and that 'the correct deformation of N=2 supersymmetric Yang-Mills theory associated to S^5 remains an open problem.' They acknowledge this, and the scaling survives across the Delta family (5.4), so the scaling claim is robust; the identification is what is conditional. Second, the 81/80 mismatch with supergravity is acknowledged, and the scheme dependence of the finite part of (4.5) is a fair defence. Third, Eq. (5.3), R_6 = g^2/(8 pi^2 k), is reverse-engineered so the quiver matches; the authors present it as a suggestion, which is honest, but it is a fit, not a derivation, and a referee should say so. Fourth, the b = 1/2 case is flagged by the authors as 'not completely valid' and should be treated as a sketch. Who gets value: people working on the 5D/(2,0) map and on localization matrix models. I would cite the coefficient and the 81/80 mismatch as a benchmark. Worth a serious referee, with the identification as the main question.

Referee report

Significance

The result is a concrete, parameter-free derivation of N^3 scaling from a 5D gauge theory, and the saddle-point algebra is elementary enough to check by hand: I verified that the virial relation following from (3.5) gives A = B/2 and hence F = −A = −27 g^2_YM N^3/(512 π r), matching (3.7), and that the supergravity finite part (4.7) follows from the expansion (4.6). The paper is transparent about the two interpretive soft spots: the computed model is N=1 SYM with an adjoint hypermultiplet rather than the maximally supersymmetric theory (§2), and the 81/80 coefficient mismatch is acknowledged and discussed (§5). The concern that the saturation of coth and tanh in passing from (3.3) to (3.4)-(3.5) is uncontrolled does not land: the corrections are exponentially small for well-separated eigenvalues, the solution (3.6) spreads the eigenvalues over O(λ), and the net correction to the O(N) force is O(N/λ), i.e. relative O(1/λ), vanishing in the strong-coupling limit. The N^3 scaling is therefore robust, and the paper's honest treatment of the open identification and the numerical mismatch is a strength rather than a defect.

Major objections (3)

[Abstract (abstract:s0); §2] The abstract states that the result is for '5D maximally supersymmetric Yang-Mills theory on S^5,' but §2 states that the computed model is N=1 SYM with an adjoint hypermultiplet (8 supercharges) and that 'the correct deformation of N=2 supersymmetric Yang-Mills theory associated to S^5 remains an open problem.' The N^3 scaling is robust across the Δ family (Eq. 5.4), so the scaling claim survives, but the identification with the maximally supersymmetric theory is not established by the calculation. The abstract and title should be qualified to match §2, for example by naming the adjoint-hypermultiplet model explicitly.
[§3, Eqs. (3.13), (3.20)-(3.21)] For b = 1/2 (M = 2N), the paper derives N^2 scaling using approximation (3.13), which it explicitly says 'is not completely valid,' and Eq. (3.21) shows that the eigenvalues spread over infinite distance as c → 0, so the finite-range argument that would justify N^2 fails precisely at this boundary. The N^2 claim at M = 2N therefore rests on an uncontrolled approximation. This is a boundary case and does not affect the M < 2N and M > 2N results, but the paper should either supply a controlled argument (for instance a numerical check of the matrix model at b = 1/2) or state the N^2 result as a heuristic expectation rather than a derived scaling.
[§5, Eq. (5.3)] The identification R_6 = g^2_YM/(8π^2 k) is introduced solely to make the quiver k-scaling of Eq. (3.11) match the supergravity result (4.8); no independent derivation or consistency check is given. Since this relation is the only way the quiver comparison is claimed to work, the paper should either justify it independently or state clearly that the quiver comparison is inconclusive without it.

Minor objections (5)

[§3, Eq. (3.6)] For ordered eigenvalues the sum Σ_{j≠i} sign(φ_i − φ_j) equals 2i − N − 1, not 2i − N. The difference is subleading in N and does not affect the leading N^3 coefficient, but the exact expression should be stated or the approximation noted.
[§3, Eq. (3.11)] The intermediate expression in Eq. (3.11) is typographically garbled in the arXiv version; the final result N^3/k^2 is clear, but the intermediate step should be cleaned up for reproducibility.
[§3, after Eq. (3.11)] The statement that 'any matrix model with such a potential will give N^3 behavior' is plausible but stated without qualification; it would be worth specifying the conditions (long-range linear repulsion and eigenvalue spread of order N) under which it holds.
[§3, Eqs. (3.14)-(3.16)] In passing from Eq. (3.14) to (3.15), dividing by φ'(x) and differentiating again assumes φ is smooth away from x = 0; the jump at x = 0 exhibited in Eq. (3.17) should be discussed explicitly.
[§5, Eq. (5.4)] The Δ-dependent coefficient (2 − Δ)^2(1 + Δ)^2/96 is quoted without derivation; a sentence indicating how it follows from the saddle-point analysis of the general Δ model would help the reader.

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The paper

The paper claims

The free energy of 5D maximally supersymmetric Yang-Mills on a five-sphere grows as the cube of the number of colors at strong coupling, matching the growth rate of its supergravity dual.

The reading

Five-dimensional maximally supersymmetric Yang-Mills theory is not a conformal theory, so putting it on a five-sphere requires a choice, but once it is there the partition function can be reduced by localization to an ordinary integral over a matrix. The authors evaluate that integral in the limit of large 't Hooft coupling and find that the eigenvalues of the matrix spread out over a range proportional to $N$, producing a free energy that scales as $N^3$ rather than the usual $N^2$. This matters because the six-dimensional $(2,0)$ theory, which has no Lagrangian description, is known from its supergravity dual to have an $N^3$ free energy, and the authors want to see how much of that behaviour survives compactification to five dimensions. They also work out the $Z_k$ quiver version and the version with matter in the fundamental representation, where the scaling drops back to $N^2$ or the model destabilizes. The headline number agrees with supergravity up to a factor of $81/80$, and the authors are candid that the residual mismatch is unresolved.

What carries the argument

The machinery is the localization matrix model for $N=1$ supersymmetric Yang-Mills on $S^5$ with one adjoint hypermultiplet, together with a special function $f(y)$ whose derivative takes the simple form $df/dy = π y^2 cot(π y)$. That identity, plus trigonometric simplification of the determinant factors, turns the exact saddle-point equation into a sum of $coth$ and $tanh$ terms weighted by $2 - (φ_i-φ_j)^2$ and $1/4 + (φ_i-φ_j)^2$. In the strong-coupling limit the eigenvalues are pushed far apart, the hyperbolic functions saturate to signs, and the equation becomes a linear force law whose solution is an equally spaced eigenvalue ladder. The same ladder structure, rescaled to blocks of $N/k$ eigenvalues, controls the quiver case, and the failure of the ladder to form controls the fundamental-matter case.

Extended reading notes

Core claim

The paper's central claim is that the perturbative partition function of 5D maximally supersymmetric Yang-Mills on $S^5$, as derived from localization, yields a free energy $F = -27 g_{YM}^2 N^3/(512 r)$ in the strong-coupling limit, where $r$ is the sphere radius. The mechanism is a long-range linear repulsion between eigenvalues: at large 't Hooft coupling the logarithm of the one-loop determinants collapses to a term proportional to $|φ_i - φ_j|$, whose saddle point places the eigenvalues at equally spaced positions $φ_i = 9λ(2i-N)/(64π^2 N)$. Because the eigenvalue cloud then has width of order $N$, both the quadratic potential term and the pairwise interaction term sum to order $N^3$. For a $Z_k$ quiver with gauge group $SU(N/k)^k$ the same computation gives $F = -27 g_{YM}^2 N^3/(512 π k^2 r)$. Adding $M$ hypermultiplets in the fundamental representation changes the scaling: for $M ≤ 2N$ the eigenvalues remain confined to a finite range and the free energy grows only as $N^2$, while for $M > 2N$ the saddle-point equations have no real solution at strong coupling and the model destabilizes. The authors compare the $N^3$ coefficient to the regularized supergravity action on $AdS_7 × S^4$, $I_{AdS} = -5g_{YM}^2 N^3/(96πr)$ under the identification $R_6 = g_{YM}^2/(8π^2)$, and find agreement in the power of $N$ but a discrepancy of $81/80$ in the coefficient.

Load-bearing premise

The derivation replaces the exact hyperbolic force between eigenvalues by a pure sign function, which is only valid if the eigenvalues end up separated by distances much larger than $1/π$; the paper verifies this only after the fact, by exhibiting the solution, rather than by bounding the discarded corrections.

Editorial extensions

If this is right

Any 5D gauge theory on $S^5$ whose localization determinant produces a linear eigenvalue repulsion will show $N^3$ free-energy scaling at strong coupling; the paper states this as a general criterion.
The $Z_k$ quiver result scales as $N^3/k^2$, which matches the orbifold supergravity answer $AdS_7 × S^4/Z_k$ only if the circle radius is identified as $R_6 = g_{YM}^2/(8π^2 k)$ rather than the naive $R_6 = g_{YM}^2/(8π^2)$.
With $M$ fundamental hypermultiplets the free energy is $N^2$ for $M ≤ 2N$ and the strong-coupling saddle point ceases to exist for $M > 2N$, giving a sharp matter-content threshold for $N^3$ behaviour.
For the family of deformed models parametrized by $Δ$, the coefficient is $-(2-Δ)^2(1+Δ)^2/96$ times $g_{YM}^2 N^3/(πr)$ for $-1 < Δ < 2$, minimized at $Δ = 1/2$, so the size of the $N^3$ effect is model-dependent.
If the identification between the Kaluza-Klein circle radius and the Yang-Mills coupling is corrected, the numerical mismatch with supergravity could be removed without changing the $N^3$ scaling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The factor $81/80$ is close enough to one that it looks like a scheme or normalization artifact rather than a structural disagreement; a natural test is whether a supersymmetric Chern-Simons term added to the 5D action shifts the coefficient by exactly that amount.
Because the finite part of the regulated supergravity action is scheme-dependent, the honest comparison is between the $N$-scaling exponents and the $k$-scaling, both of which the matrix model reproduces; the coefficient comparison may be over-weighted in the paper's own discussion.
The threshold $M = 2N$ resembles a deconfinement-type transition in the eigenvalue density, and one could look for it directly as a change in the shape of the eigenvalue distribution at finite $M/N$ rather than only in the free-energy exponent.
If the linear-repulsion criterion is right, it gives a quick diagnostic for any localized gauge theory: inspect the strong-coupling limit of the one-loop determinant for a $|φ_i - φ_j|$ term before doing any saddle-point work.

Original abstract

In this note we derive $N^3$-behavior at large t' Hooft coupling for the free energy of 5D maximally supersymmetric Yang-Mills theory on $S^5$. We also consider a $Z_k$ quiver of this model, as well as a model with $M$ hypermultiplets in the fundamental representation. We compare the results to the supergravity description and comment on their relation.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.69 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

In this note we derive $N^3$-behavior at large t' Hooft coupling for the free energy of 5D maximally supersymmetric Yang-Mills theory on $S^5$. We also consider a $Z_k$ quiver of this model, as well as a model with $M$ hypermultiplets in the fundamental representation. We compare the results to the supergravity description and comment on their relation.

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s0	In this note we derive $N^3$-behavior at large t' Hooft coupling for the free energy of 5D maximally supersymmetric Yang-Mills theory on $S^5$.	result 0.61	0.97	0.93	asserted 1.00
s1	We also consider a $Z_k$ quiver of this model, as well as a model with $M$ hypermultiplets in the fundamental representation.	contribution 1.00	0.89	0.03	asserted 0.81
s2	We compare the results to the supergravity description and comment on their relation.	interpretation 0.51	0.83	0.05	unclear 0.63

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References

Works cited (25)

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Forward citations

Cited by 6 Pith papers

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

Superconformal Partition Functions and Non-perturbative Topological Strings
hep-th 2012-10 conditional novelty 7.0 of 10

The non-perturbative refined topological string is defined as a triple product of perturbative amplitudes with modular-inverted couplings, and this reproduces the known single-M5-brane index.
Exact results for five-dimensional superconformal field theories with gravity duals
hep-th 2012-07 accept novelty 7.0 of 10

The S^5 free energy of five-dimensional superconformal theories with AdS_6 duals is computed exactly by localization and matches the holographic entanglement entropy, confirming the N^{5/2} scaling.
Towards Classification of 5d SCFTs: Single Gauge Node
hep-th 2017-05 conditional novelty 6.0 of 10

A relaxed Coulomb-branch consistency criterion, plus three conjectured equivalents, yields a table-by-table classification of 5d gauge theories with a single simple gauge group that could have interacting fixed points.
Supersymmetric Casimir Energy and the Anomaly Polynomial
hep-th 2015-07 conditional novelty 6.0 of 10

The supersymmetric Casimir energy of an even-dimensional superconformal field theory equals the equivariant integral of its anomaly polynomial, checked in two, four and six dimensions.
Perturbative partition function for a squashed S^5
hep-th 2012-10 conditional novelty 6.0 of 10

The perturbative partition function of 5d supersymmetric gauge theories on a squashed five-sphere is derived and written as a product of triple sine functions whose periods are set by the squashing parameters.

Show 1 more citing papers

The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere
hep-th 2012-06 conditional novelty 6.0 of 10

The full perturbative partition function of 5D supersymmetric Yang-Mills with matter on S5 is computed by localization and expressed as a matrix integral over the Cartan subalgebra.

How later papers used it

71 later papers cite it

Evidence: substantial. 58 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 24 of the 111 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 1 paper · 1 by the same authors

By the same authors

Followed this paper's analysis to get the free energy at large N for the vector multiplet coupled to an adjoint hypermultiplet with mass m on a squashed toric SE manifold.

Gluing Nekrasov partition functions (2014) (same authors) · 0.87 · paraphrased: the licence does not allow quoting

Compares with it 1 of 11 papers with this label

Found that its derivation in the M-theory limit produces the same result as this paper's derivation, that is, first take the 't Hooft limit and then go to strong coupling.

Large $N$ limits of supersymmetric quantum field theories: A pedagogical overview (2025) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

71 arXiv papers cite this in 111 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 70 sentences without a confident label

Builds on it 2 papers

The results for the free-energy and the Wilson loop can be generalized to a $ℤ_k$ quiver of the $𝒩=1^*$ theory [ref] .

Matrix models for 5d super Yang-Mills · $𝒩=1^*$ $5d$ super Yang-Mills · 0.97
The above results can be straightforwardly generalized to a $ℤ_k$ quiver of the $=1^*$ theory [ref] .

5D super Yang-Mills theory and the correspondence to AdS$_7$/CFT$_6$ · Quivers · 1.00

Uses its method or tool 1 paper

Using these asymptotics and following the analysis from [ref] we get the free energy at the large $N$ limit for the vector multiplet coupled to a hypermultiplet in adjoint with mass $m$ F = - logZ = - ^2 N^3/96 πr ϱ( 1/4 (^1)^2 + m^2 )^2 .for a squashed toric SE manifold.

Gluing Nekrasov partition functions · The Asymptotic Behaviour and Large $N$ · 0.87

Uses its data or values 1 paper

Using localization, the $=1^*$ free energy at strong coupling is found to be [ref] F=-(9/4+m^2)^2/96 π N^2 , where $=g_YM^2N/r$ is the 't Hooft parameter and $m≡i rM$.

5D super Yang-Mills theory and the correspondence to AdS$_7$/CFT$_6$ · Introduction and main results · 0.95

Compares with it 11 papers

In this section we compare the results for strongly coupled 5d SYM with the corresponding results found using the AdS/CFT correspondence [ref] .

Matrix models for 5d super Yang-Mills · Supergravity comparisons · 1.00
To start we review the supergravity computation of the free energy given in [ref] .

5D super Yang-Mills theory and the correspondence to AdS$_7$/CFT$_6$ · Supergravity comparisons · 0.89
The partition function of $ℂℙ^2×ℝ$ QFT reproduces the `Casimir energy' which exhibits an $N^3$ scaling at large $N$, which was first obtained from the QFT on $S^5$ [ref] .

The general M5-brane superconformal index · Introduction · 0.95

Cites it as background 24 papers

The 6d superconformal index was studied from localization of 5d MSYM on $S^5$ [ref] .

Bootstrapping ADE M-strings · (2,0) Superconformal index and $𝒲$-algebra · 0.82
We begin by reviewing the supergravity computation of the free energy [ref] .

Matrix models for 5d super Yang-Mills · Supergravity comparisons · 0.89
See [ref] for studies on the off-shell classical field theory on round $S^5$, [ref] for the studies of the theories on squashed $S^5$, [ref] for the studies on perturbative part of the partition function, [ref] for the studies on the non-perturbative corrections, [ref] for $N^3$ scalings of the index version of 6d Casimir energies, [ref] for BPS Wilson loops on $S^5$ which are interpreted as Wilson surfaces in 6d.[ 6d $(2,0)$ partition function on other manifolds with $S^1$ factors was also studied from 5d SYM.

The general M5-brane superconformal index · Introduction · 1.00

Discussion

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[1] [12]

J. K¨ all´ en, J. Qiu, and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the ﬁve-sphere , JHEP 1208 (2012) 157, [ arXiv:1206.6008]

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