REVIEW
Rigorous Test of Non-conformal Holography: Wilson Loops in N=2* Theory
High Energy Physics - Theory · Submitted 8 Jan 2013 · 14 references
Pith's verdict: A clean, short, and genuinely two-sided check of holography in N=2*: localization reproduces the Pilch-Warner vacuum width, the perimeter law, and the leading free energy, with only cosmetic soft spots.
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Referee report
The paper
The paper claims
A localization computation at large $N$ and strong 't Hooft coupling reproduces, in $N=2^*$ gauge theory, both the Coulomb-branch eigenvalue distribution and the Wilson-loop expectation value predicted by the Pilch-Warner supergravity dual.
The reading
What carries the argument
The carrying object is the localization matrix integral on $S^4$, whose integrand is a product of pairwise interactions built from a single function $H(x)$ (a regularized infinite product) plus a Gaussian weight $2$. Its large-$N$ saddle-point equation is a singular integral equation for the eigenvalue density, with kernel $2$. The load-bearing step is the Taylor expansion $2$, which converts the equation into the standard Wigner form solved by a semicircle; the Wilson loop is then the exponential average $2$, dominated by the largest eigenvalue. On the gravity side the corresponding machinery is the Nambu-Goto action for a string ending on the contour in the warped metric $2$, renormalized by subtracting the perimeter divergence.
Extended reading notes
Core claim
The paper's central claim is that the localization matrix model of $N=2^*$ theory, taken at large $N$ and large 't Hooft coupling, reproduces two independent supergravity predictions for the Pilch-Warner background. First, the saddle-point eigenvalue density is Wigner's semicircle with width $2$ in units of the four-sphere radius, $2$, which becomes $2$ when the sphere is decompactified — the same width read off from the supergravity solution. Second, the same saddle point gives $2$, so that at large radius $2$: a perimeter law, not an area law, whose coefficient matches the renormalized minimal-surface area in the Pilch-Warner metric, where the complicated factor $2$ cancels between the induced metric and the dilaton. The authors also compute the free energy at strong coupling, $2$, which at large radius behaves as $2$.
Load-bearing premise
The derivation replaces the exact interaction kernel by its second-order Taylor term, $2$, which is legitimate only when the symmetry-breaking scale $2$ is much larger than the deformation mass $M$; the paper assumes this before solving, confirms it only afterwards from its own answer, and supports it with numerical tests that are asserted in a footnote but not displayed.
Editorial extensions
If this is right
- At strong coupling the eigenvalue density is semicircular with $2$, so on $2$ the symmetry-breaking scale is $2$, parametrically larger than the deformation mass $M$ by a factor of order $2$.
- Large Wilson loops obey $2$ rather than an area law, in both the localization computation and the minimal-surface computation, with coinciding coefficients.
- The strong-coupling free energy is $2$, reducing to $2$ at large radius with logarithmic accuracy.
- The agreement holds at finite four-sphere radius $R$ as well, so the results $2$, $2$ and $2$ are predictions that go beyond the decompactified supergravity limit.
Reading between the lines
- Because the reduction to a Gaussian matrix model only uses the semicircular form of the kernel at strong coupling, the same route should apply to other $2$ quiver theories whose gravity duals exhibit semicircular eigenvalue densities; checking whether the free-energy formula survives there is a concrete next test.
- The finite-$R$ results are predictions for a half-BPS supergravity solution whose boundary is $S^4$ rather than $2$; constructing that solution would independently test the $2$ shift in the eigenvalue width.
- The subleading correction to the circular loop quoted in the appendix, $2$, is a sharp target: a numerical string computation at large but finite loop radius, or a localization computation at finite $2$, could confirm or contradict it.
Original abstract
N=2* gauge theory in four space-time dimensions arises as a deformation of the parent N=4 supersymmetric Yang-Mill theory under which its hypermultiplet acquires a mass. This theory, on the one hand, has a known supergravity dual and on the other hand is amenable to localization. We explain, using localization, a dynamical selection of the supergravity Coulomb branch vacuum at large N. We also demonstrate that large Wilson loops obey perimeter law, again finding exact match between direct field-theory calculations and string theory. Additionally, we compute free energy at large 't Hooft coupling.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
N=2* gauge theory in four space-time dimensions arises as a deformation of the parent N=4 supersymmetric Yang-Mill theory under which its hypermultiplet acquires a mass. This theory, on the one hand, has a known supergravity dual and on the other hand is amenable to localization. We explain, using localization, a dynamical selection of the supergravity Coulomb branch vacuum at large N. We also demonstrate that large Wilson loops obey perimeter law, again finding exact match between direct field-theory calculations and string theory. Additionally, we compute free energy at large 't Hooft coupling.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | N=2* gauge theory in four space-time dimensions arises as a deformation of the parent N=4 supersymmetric Yang-Mill theory under which its hypermultiplet acquires a mass. | background 1.00 | 0.05 | 0.03 | asserted 1.00 |
| s1 | This theory, on the one hand, has a known supergravity dual and on the other hand is amenable to localization. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s2 | We explain, using localization, a dynamical selection of the supergravity Coulomb branch vacuum at large N. | contribution 0.69 | 0.95 | 0.25 | asserted 1.00 |
| s3 | We also demonstrate that large Wilson loops obey perimeter law, again finding exact match between direct field-theory calculations and string theory. | result 0.99 | 0.96 | 0.41 | asserted 1.00 |
| s4 | Additionally, we compute free energy at large 't Hooft coupling. | result 0.62 | 0.96 | 0.08 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:1301.1597 by the authors.
References
How the literature uses it
64 arXiv papers cite this in 153 sentences
Papers citing it · 103 sentences without a confident label
Contests it 1 paper
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The dashed red curve represents the leading approximation to $T_DFP$ in the limit of small conformal symmetry breaking $m/H→0$, see[The $1/2$ scaling exponent is correct, but the prefactor must be modified to account for the fact that the bulk scalar dual to $_3$ does not have a canonical kinetic term; furthermore, the precision holography in this case requires that $m∝Λ$, but is not equal to it [ref] . ] [ref].
Gravitational reheating at strong coupling · Reheating in $=2^*$ gauge theory · contest 0.71
Builds on it 1 paper
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Our holographic result is valid for general values of the parameter $ma$ and thus provides a generalization of the analysis in [ref] .
Precision Holography for $\mathcal{N}=2^{*}$ on $S^4$ from type IIB Supergravity · Introduction · 0.97
Uses its method or tool 6 papers
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We will follow the procedure outlined in [ref] to obtain the large $N$, $λ$ limit of the localization result of $SU(N)$ $N=$4 SYM on the ellipsoid obtained in section [ref].
Super-R\'enyi Entropy & Wilson Loops for N=4 SYM and their Gravity Duals · The Partition Function at large $N$ · 0.99
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The main reason for focusing on these derivatives of $Z(b, m, τ, τ̅)$ is that $Z(b, m, τ, τ̅)$ itself can be computed exactly at any $N$ and any coupling $(τ, τ̅)$ using supersymmetric localization [ref] (see also [ref] ).
New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory · Introduction · 0.85
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Owing to the ability to apply localization to the theory on $S^4$ [ref] , the strong-coupling behaviour of the free energy, Wilson loops [ref] , and the theory's phases [ref] have been explored using matrix model techniques.
Holographic Dual of the Eguchi-Kawai Mechanism · Introduction · 0.83
Uses its data or values 2 papers
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This function is known in the two limiting cases [ref] : [equation] where $γ$ is the Euler constant.
Evidence for Large-N Phase Transitions in N=2* Theory · $𝒩=2^*$ theory · 0.83
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A more accurate calculation that keeps track of the $λ$-independent constant gives [ref] : [equation]
Localization at Large N · $𝒩=2^*$ theory · 0.83
Compares with it 10 papers
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The result for the third derivative of the free energy $F$ with respect to the mass is in perfect agreement with the field theory calculation in [ref] , which used the matrix integral arising from the path integral localization formula of Pestun [ref] to compute the partition function of the theory.
Holography for ${\cal N}=2^*$ on $S^4$ · Discussion · 1.00
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The strong-coupling interpolation of this result perfectly matches with the first term in ([ref]) [ref] .
Holographic Dual of the Eguchi-Kawai Mechanism · Strong coupling · 0.97
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From this solution we can calculate the expectation value of the Wilson loop ([ref]): [equation] The leading exponential behavior corresponds to the minimal area law in the dual gravitational description, and at $M→∞$ one finds a precise match with the area law in the Pilch-Warner background [ref] .
N=2* Super-Yang-Mills Theory at Strong Coupling · Localization · 0.87
Cites it as support 2 papers
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This follows from the supergravity analysis of the known holographic dual [ref] of $𝒩=2^*$ SYM [ref] , or can be derived directly from field theory using localization [ref] .
Massive N=2 Gauge Theories at Large N · $𝒩=2^*$ SYM · 0.98
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This assertion is based on universality and has less rigorous grounds compared to localization, but has been checked holographically for Wilson loops in the fundamental representation [ref] .
Higher Rank Wilson Loops in N = 2* Super-Yang-Mills Theory · Wilson loops · 0.98
Cites it as background 21 papers
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The phase structure of the theory at $Q=0$ was understood in [ref] .
Phase structure of $\mathcal{N}=2^*$ SYM on ellipsoids · $Q-$driven phase transitions · 0.87
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This equation was studied in a number of papers [ref] , but its general analytic solution is not known[An equation with a similar kernel arises in the $𝒩=4$ SYM compactified on the squashed four-sphere [ref] .
Strong-Coupling Phases of Planar N=2* Super-Yang-Mills Theory · Localization matrix model · 0.88
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This dramatic simplification makes exact calculations possible and has led to a large volume of work studying its application in the context of AdS/CFT [ref] .
A Precision Test of AdS/CFT with Flavor · Introduction · 0.92
What it cites · 11 sentences without a confident label
Uses the method or tool of 3 papers
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We will consider $𝒩=2^*$ theory on the four-sphere of radius $R$, which here we regard merely as an IR regularization, and will use localization results [ref] to analyze the theory at large $N$ in the regime of the strong 't Hooft coupling.
Localization of gauge theory on a four-sphere and supersymmetric Wilson loops · Introduction · 1.00
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The metric ([ref]) is written in the Einstein frame and is normalized such that the string action is given by [equation] An expectation value of the Wilson loop, at strong coupling, is determined by the renormalized on-shell action of the string that ends on the contour $C$ at the boundary of AdS [ref] : [equation] Renormalization consists in integrating from the cutoff surface at $c_ϵ=1+ϵ^2M^2/2$ and subtracting the perimeter divergence: [equation] Here we used the relationship ([ref]) between $c$ and the radial coordinate in $AdS_5$ in the asymptotic region near the boundary.
Macroscopic strings as heavy quarks: Large-N gauge theory and anti-de Sitter supergravity · Strong coupling from supergravity · 0.82
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The metric ([ref]) is written in the Einstein frame and is normalized such that the string action is given by [equation] An expectation value of the Wilson loop, at strong coupling, is determined by the renormalized on-shell action of the string that ends on the contour $C$ at the boundary of AdS [ref] : [equation] Renormalization consists in integrating from the cutoff surface at $c_ϵ=1+ϵ^2M^2/2$ and subtracting the perimeter divergence: [equation] Here we used the relationship ([ref]) between $c$ and the radial coordinate in $AdS_5$ in the asymptotic region near the boundary.
Wilson loops in large N field theories · Strong coupling from supergravity · 0.82
Compares with 2 papers
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At strong coupling, it grows as $e ^√(λ)$, in precise agreement with the area law for classical strings in $AdS_5×S^5$ [ref] .
Wilson Loops in N=4 Supersymmetric Yang--Mills Theory · Strong coupling from localization · 0.93
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At strong coupling, it grows as $e ^√(λ)$, in precise agreement with the area law for classical strings in $AdS_5×S^5$ [ref] .
An Exact Prediction of N=4 SUSYM Theory for String Theory · Strong coupling from localization · 0.93
Cites as background 11 papers
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The gravity dual of this theory is an explicitly known solution of ten-dimensional supergravity [ref] .
N=2 Supersymmetric RG Flows and the IIB Dilaton · Introduction · 0.86
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The study of the matrix integral ([ref]) at large-$N$ was initiated in [ref] , following earlier work on similar models [ref] .
Wilson Loops in N=2 Super-Yang-Mills from Matrix Model · Strong coupling from localization · 0.82
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The study of the matrix integral ([ref]) at large-$N$ was initiated in [ref] , following earlier work on similar models [ref] .
A note on the integral equation for the Wilson loop in N = 2 D=4 superconformal Yang-Mills theory · Strong coupling from localization · 0.82
Discussion
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