Higher rank Wilson loops from a matrix model

Sean A. Hartnoll; S. Prem Kumar

REVIEW

Higher rank Wilson loops from a matrix model

Sean A. Hartnoll, S. Prem Kumar

High Energy Physics - Theory · Submitted 3 May 2006 · 25 references

Pith's verdict: A clean, honest consolidation: one contour integral reproduces both the D3 and D5 brane answers, with the second symmetric saddle as the one genuinely under-supported piece.

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arXiv IDhep-th/0605027 (version 2)

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CitationSean A. Hartnoll, S. Prem Kumar. "Higher rank Wilson loops from a matrix model." arXiv:hep-th/0605027 (2006). https://pith.science/paper/NHRADRMO

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The thing to know: this paper puts the D3-brane and D5-brane supergravity answers for circular Wilson loops inside a single matrix-model contour integral, and the leading strong-coupling reductions check out. I went through both by hand. The antisymmetric saddle equation gives z = cos θ_k with π(1−f) = θ_k − sinθ_k cosθ_k and the exponent lands on (2N/3π)√λ sin³θ_k, matching Yamaguchi's D5 result. The symmetric saddle y1 = −√(1+κ²) solves (2.40) and gives 2N(κ√(1+κ²) + sinh⁻¹κ), matching Drukker-Fiol, and reducing to exp[√λ] at k=1. That is real cross-checking of the representation dictionary, not a tautology: the brane computations are independent inputs from [13] and [19]. The weak-coupling Casimir formula (2.51) is a nice bonus and looks right. The critical-coupling curve in figure 1 is a genuinely new observation about where the symmetric saddle leaves the first sheet, and it is the most interesting thing in the paper even though the authors do not push it far. The load-bearing premise is (2.3), the Drukker-Gross conjecture extended to arbitrary representations. They state it as a conjecture and then test it, which is the correct posture; I would not call that a flaw. Where I would push back is W2, equation (2.47). The saddle is found by taking λ→∞ in the exponent first and then solving (2.44), which is not obviously the same as a saddle of the full exponent, the claimed positivity is asserted rather than shown, and writing W_S = W1 + W2 presumes both saddles sit on the same deformed contour without demonstration. None of this touches the D3 match, since W1 dominates in the limits used, but the W2 discussion reads as more settled than it is. The factor-of-4 argument around (2.40) is compressed; a reader can reconstruct it, but it takes work. Dropped phases in the exponents are fine at leading order and flagged. The overlap note with [18] is disclosed and does not undercut anything. This is a paper for anyone working on higher-representation Wilson loops or the brane-representation dictionary; it is the reference that ties the two brane results together and gives the finite-coupling corrections. It deserves a serious referee, and I would send it out without hesitation.

Referee report

Significance

The result is a genuine and parameter-free cross-check of the holographic dictionary for Wilson loops in arbitrary representations proposed by Gomis and Passerini: two independent brane computations, the D3 result of Drukker-Fiol and the D5 result of Yamaguchi, are recovered from one matrix-model contour integral in the appropriate limits, with no adjustable constants. The framework also resolves the earlier confusion about why a multiply wound loop matched a D3-brane computation, and it supplies computable e^{-√λ} corrections that delineate the regime of validity of the supergravity approximations. The weak-coupling formula (2.51) is a falsifiable prediction that can be tested against perturbation theory for mixed representations, and the identification of λ_c(f) is a concrete, checkable statement about where the symmetric-representation saddle structure changes. The load-bearing input, Eq. (2.3), is openly stated as the Drukker-Gross conjecture extended to arbitrary representations and is then tested rather than assumed silently, which is the right way to proceed.

Major objections (1)

[§2.5, Eqs. (2.39)-(2.48)] The second symmetric saddle W₂ and the decomposition W_S = W₁ + W₂ are the least supported part of the paper. Three steps need justification. First, W₂ is obtained by taking λ → ∞ inside the exponent of (2.39) and only then solving the saddle equation (2.44); it is not shown that this commutes with finding the stationary point of the full exponent, and the resulting φ_k is complex, so the usual steepest-descent bookkeeping does not automatically apply. Second, the positivity of the exponent at that saddle is asserted (the text says 'we have checked numerically') but no values, ranges of f and λ, or figure are given, so the reader cannot verify that W₂ is non-negligible in the large-N limit. Third, and most importantly, writing the Wilson loop as the sum of two stationary-point values presupposes that both saddles lie on the same contour obtained by deforming the original contour of (2.26) around the branch cut; the deformation itself deserves a paragraph, because the original contour sits at t = 0 (Re z → -∞), which is an essential singularity of the integrand for the symmetric case, and the saddle y₁ = -√(1+κ²) lies between that point and the cut. The claim in the text that 'for general values of λ and f, either saddle could dominate' rests on all three points. This does not affect the D3 match, which uses W₁ alone in the regime where it dominates, but it does affect the stated result (2.48) and the discussion in the conclusions, and it should either be supported or explicitly downgraded to a conjecture.

Minor objections (11)

[§2.5, Eq. (2.38)] The large-λ_c coefficient appears to be off by a factor of √π. From (2.37), setting a = √λ_c and u = x+1, one has f = -(2/π)∫₀² du √(u(2-u))/(1-e^{au}); for large a this gives f ≈ (2/π)√2 ∫₀^∞ u^{1/2}/(e^{au}-1) du = (2/π)√2 Γ(3/2)ζ(3/2) a^{-3/2} = √(2/π) ζ(3/2) λ_c^{-3/4} ≈ 2.08 λ_c^{-3/4}, whereas the manuscript quotes √2 ζ(3/2)/π λ_c^{-3/4} ≈ 1.18 λ_c^{-3/4}. The small-λ_c limit f ≃ 2/√λ_c I do confirm. Please check the coefficient; the qualitative conclusion that the saddle disappears is unaffected.
[§2.5, Eqs. (2.28)-(2.32)] The imaginary part of the antisymmetric saddle equation gives sin(√λ Im z) = 0, whose general solution is Im z = nπ/√λ, and the n = 1 image lies exactly on the branch cut. The statement that 'only the solution Im z = 0 (and images) is picked out' is asserted rather than shown; a sentence identifying which images lie on the relevant contour, or are endpoints of it, would make the saddle selection convincing.
[§2.5, Eq. (2.40)] The factor of 4 in (2.40) is attributed to analytic continuation of the logarithm to large negative y, but the continuation is not written out. Since this factor fixes the coefficient 2 in the D3 result (2.43), an explicit display of the continued exponent, showing where the doubling of the third integral comes from, would be worth including.
[§2.5, text following Eq. (2.31)] The sentence 'It is clear that in the regions Re z > 1 and Re z < 1 there exist no consistent solutions' should read 'Re z > 1 and Re z < -1'; as written the two regions cover the whole plane and contradict the following paragraph.
[§2.5, Eqs. (2.43) and (2.47)] Both results are quoted 'ignoring phases'. For a Wilson-loop expectation value that should be real and positive, and for the sum W₁ + W₂ in particular, the relative phase of the two contributions matters for any interference; a sentence explaining why the phases can be dropped, or how they would enter, would be helpful.
[§2.5, Eq. (2.36) and the k → N-k discussion] The invariance of (2.36) under θ_k → π - θ_k is stated as a property of the integral (2.26). The simpler reason is that A_k and A_{N-k} are complex conjugate representations and the BPS circular loop expectation is real; stating this would make the symmetry transparent.
[§2.5, small-f limit] It may be worth noting that for small f both (2.36) and (2.43) reduce to exp[k√λ], the multiply wound fundamental-string answer, since sin³θ_k ≈ 3πf/2 and 2N(κ√(1+κ²) + sinh⁻¹κ) ≈ 4Nκ = k√λ. This connects directly to the puzzle raised in the introduction about the k-winding computation and would strengthen the discussion there.
[§2.6, Eq. (2.51)] The formula exp[λC₂(R)/(8N)] is derived for the two single-row/column cases and then proposed for arbitrary mixed representations. Since this is presented as a prediction, it would be worth either checking one mixed representation explicitly or stating more explicitly that the generalization is conjectural.
[Footnote 2] The footnote states that the method of [19] 'differs from the correct finite λ result'. Since [19] was computing the leading large-λ answer, this comparison should be phrased more carefully, for instance by saying that the cumulative-distribution method captures the leading large-λ exponent but not the e^{-√λ} corrections.
[Abstract and §2.1] The abstract's 'expressions for all values of the 't Hooft coupling' refers to the large-N contour integral (2.26), which is exact in λ but not in 1/N; a few words making the large-N qualification explicit would avoid over-reading.
[§2.4, Eq. (2.26)] The description of the contour as lying 'to the left of the branch cut, in the region Re z < -1' and winding once around the z-plane cylinder would benefit from one sentence fixing the cylinder coordinate and stating that the contour encircles t = 0, since this is what makes the residue prescription of §2.3 and the deformation of §2.5 consistent.

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The paper

The paper claims

One Gaussian matrix integral reproduces both the D3- and D5-brane predictions for high-rank circular Wilson loops.

The reading

The paper asks whether one Gaussian Hermitian matrix model can compute circular BPS Wilson loops of $mathcal{N}=4$ super Yang-Mills in high-rank representations, not only the fundamental one. A rank-$k$ symmetric or antisymmetric representation is built by symmetrizing or antisymmetrizing $k$ copies of the fundamental, and the authors keep $f=k/N$ fixed as $N$ grows. Taking the conjectured exact matrix-model answer for circular BPS loops in any representation, they turn the rank-$k$ trace into a contour integral and evaluate it by saddle point at leading large $N$ for all values of the 't Hooft coupling $lambda$. At strong coupling the antisymmetric integral reproduces the D5-brane result and the symmetric integral reproduces the D3-brane result, while at weak coupling both reduce to an exponential of the quadratic Casimir, $exp[lambda C_2(R)/(8N)]$. The result gives one framework for these brane matches and for corrections suppressed by powers of $e^{-sqrt{lambda}}$.

What carries the argument

The machinery is a contour-integral encoding of the rank-$k$ traces. For the antisymmetric representation the $k$-th elementary symmetric polynomial of $e^{m_i}$ is the coefficient of $t^{N-k}$ in $mathrm{det}(t+e^M)$; for the symmetric representation the complete homogeneous polynomial is the coefficient of $t^{-k-1}$ in $1/mathrm{det}(1-t e^M)$. Inserting these generating functions into the Gaussian matrix integral and using the semicircle eigenvalue density turns each Wilson loop into a single integral over $z=t e^{-sqrt{lambda}}$. The large-$N$ saddle point of that integral reproduces the brane actions; for symmetric representations the contour must be deformed around a branch cut, which produces the two saddles and the critical coupling where the negative-real saddle disappears.

Extended reading notes

Core claim

The central claim is that the large-$N$ circular BPS Wilson loop in the rank-$k$ symmetric or antisymmetric tensor representation is captured by a single contour integral, $$langle mathrm{Tr}_{S_k,A_k}[U] rangle = frac{sqrt{$\lambda$}}{2pi i} int_C dz, exp!left[ mp Nleft( frac{2}{pi}int_{-1}^{1}dx,sqrt{1-$x^{2}$}log(1 mp $e^{{-sqrt{lambda}}$(x-z)}) pm fsqrt{$\lambda$}z right) right],$$ with $f=k/N$ and the upper signs for the symmetric case. This integral follows from the Gaussian matrix model at leading large $N$ and is exact in $lambda$; its saddle points give the strong-coupling D3 and D5 brane answers and the weak-coupling Casimir formula. The antisymmetric saddle satisfies $pi(1-f)=theta_k-sin theta_k cos theta_k$ and yields $W_{A_k}=exp[(2N/3pi)sqrt{lambda}sin^3 theta_k]$. The symmetric case has two competing saddles, one reproducing the D3-brane result $W_1=exp[2N(kappasqrt{1+kappa^2}+mathrm{sinh}^{-1}kappa)]$ with $kappa=fsqrt{lambda}/4$, and another whose location is $lambda$-independent in the strong-coupling limit.

Load-bearing premise

The load-bearing premise is that the Gaussian Hermitian matrix model computes the circular BPS Wilson loop in every representation, not only the fundamental one; the paper assumes this and then checks it against brane results.

Editorial extensions

If this is right

The D3-brane with $k$ units of worldvolume flux is dual to the rank-$k$ symmetric representation, while the D5-brane is dual to the rank-$k$ antisymmetric representation, as proposed by the general Wilson-loop dictionary.
The strong-coupling Wilson loops contain corrections of order $e^{-sqrt{lambda}}$ relative to the supergravity values, so the brane calculations are the leading term in a systematically computable expansion.
At weak coupling the loops take the form $exp[lambda C_2(R)/(8N)]$, which the authors expect to hold for arbitrary tensor representations with mixed symmetry.
For antisymmetric representations the rank-$k$ and rank-$(N-k)$ loops coincide, following from $theta_k to pi-theta_k$.
For symmetric representations the existence of the negative-real saddle depends on $f$ and $lambda$, with a critical curve separating a saddle region from a no-saddle region.

Reading between the lines

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

If the arbitrary-representation matrix-model premise holds, the same contour method should extend to mixed-symmetry representations and to finite-temperature Polyakov loops up to an overall normalization, as the D5 black-hole comparison hints.
The critical coupling at which the symmetric saddle leaves the first sheet may show up as a non-analytic feature in the large-$N$ loop; looking for a matrix-model phase transition in $W_{S_k}(lambda)$ would test that reading.
The $e^{-sqrt{lambda}}$ corrections are a natural target for a one-loop fluctuation calculation around the D3 and D5 probe branes, giving a precision check beyond the leading supergravity match.
The weak-coupling Casimir formula can be checked against ordinary perturbative gauge theory at finite $k/N$, independent of the matrix-model conjecture.

Original abstract

We compute the circular Wilson loop of N=4 SYM theory at large N in the rank k symmetric and antisymmetric tensor representations. Using a quadratic Hermitian matrix model we obtain expressions for all values of the 't Hooft coupling. At large and small couplings we give explicit formulae and reproduce supergravity results from both D3 and D5 branes within a systematic framework.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We compute the circular Wilson loop of N=4 SYM theory at large N in the rank k symmetric and antisymmetric tensor representations. Using a quadratic Hermitian matrix model we obtain expressions for all values of the 't Hooft coupling. At large and small couplings we give explicit formulae and reproduce supergravity results from both D3 and D5 branes within a systematic framework.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We compute the circular Wilson loop of N=4 SYM theory at large N in the rank k symmetric and antisymmetric tensor representations.	contribution 0.99	0.95	0.20	asserted 1.00
s1	Using a quadratic Hermitian matrix model we obtain expressions for all values of the 't Hooft coupling.	method 1.00	0.94	0.12	asserted 1.00
s2	At large and small couplings we give explicit formulae and reproduce supergravity results from both D3 and D5 branes within a systematic framework.	result 0.96	0.93	0.33	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.91.

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Figures

**Figure 1.** The existence of a saddle point at negative real z depends on f and λ as shown. The asymptotics of the curve in figure 1 may be derived analytically. We have f ≃ 2 λ 1/2 c + · · · as λc → 0 , f ≃ r 2 π ζ( 3 2 ) λ 3/4 c + · · · as λc → ∞ . (2.38) In the region where the saddle point exists, we can compute the location of the saddle point numerically and evaluate (2.26) at the saddle, thereby giving us the Wilson loop expectation value. However, at any fixed f, once the coupling is sufficiently strong there are no saddle points on the first sheet of the complex plane. W

Figures reproduced from arXiv:hep-th/0605027 by the authors.

References

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55 later papers cite it

Evidence: substantial. 51 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 119 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

4 later papers by other authors are listed below: 2 built on it, 1 used its method or tool and 1 compared its own results with it.

Builds on it 2 of 3 papers with this label

Extended this paper's derivation, which does not depend on the specific theory as long as the Wilson loop vev is localized to a finite-dimensional integral, to the five-dimensional setting with a few improvements to accommodate a non-even eigenvalue density.

Phases of five-dimensional supersymmetric gauge theories (2021) · 0.91 · paraphrased: the licence does not allow quoting
Generalized the methods used in this paper.

A prediction for bubbling geometries (2007) · 0.83 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 5 papers with this label

Adopted the technique described in this paper to find the explicit form of the Wilson loop.

Phase transitions and Wilson loops in antisymmetric representations in Chern-Simons-matter theory (2018) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 7 papers with this label

Compared with this paper's result by performing the integration by parts and changing the variable.

Giant Wilson Loops and AdS$_2$/dCFT$_1$ (2020) · 0.94 · paraphrased: the licence does not allow quoting

How the literature uses it

52 arXiv papers cite this in 109 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 · 68 sentences without a confident label

Contests it 1 paper

Therefore, the calculation in [ref] needs to be re-interpreted.

Comments on Higher Rank Wilson Loops in ${\cal N}=2^*$ · Further remarks on the antisymmetric representation · contest 0.84

Builds on it 3 papers

In the gauge theory, the computation of these correlation functions makes use of a matrix model (the computation of the correlators when the Wilson loop is in the fundamental, symmetric and antisymmetric representation was performed in [ref] ).[For the corresponding Wilson loop expectation value computation see [ref] .] Building on the results in [ref] , we solve the matrix model capturing these correlation functions in the strong coupling regime and when the Wilson loop is in a large representation.

Wilson loop correlators at strong coupling: from matrices to bubbling geometries · Introduction and outline · 0.89
The derivation of [ref] does not depend on the specific theory, as long as the Wilson loop vev is localized to a finite-dimensional integral, and directly extends to the present five-dimensional setting, with a few improvements to accommodate a non-even eigenvalue density.

Phases of five-dimensional supersymmetric gauge theories · Wilson loops in large antisymmetric representations · 0.91
The techniques we introduce here are a generalization of the methods used in [ref] .

A prediction for bubbling geometries · Rectangular Young diagram · 0.83

Uses its method or tool 4 papers

Then, in the continuum limit, when the integral is dominated by the saddle point densities [ref][ref] we can write [ref] [equation] where the upper (lower) sign is for the symmetric (antisymmetric) representation.

Higher rank Wilson loops in the $\mathcal{N} = 2$ $SU(N)\times SU(N)$ conformal quiver · Higher rank Wilson loops · 0.92
According to the features exposed at the beginning of Section [ref], the method developed in [ref] allows to obtain the Wilson loop in the antisymmetric representation of rank $k$ through [ref]: [equation] where we recall that $f:= k /N$ is kept finite at large $N$, and $γ$ is a circle of radius $1/AR$.

Phase transitions and Wilson loops in antisymmetric representations in Chern-Simons-matter theory · Wilson loop with insertion of FI term · 0.98
Following [ref] we arrive at [equation] in the large $N$ approximation.

Phases of five-dimensional supersymmetric gauge theories · Wilson loops in large antisymmetric representations · 0.93

Compares with it 7 papers

We, therefore, suggest the following saddle-point configuration: the eigenvalues $m_i$ obey the Wigner distribution and $u_1$ is inserted into an equilibrium position, i.e. [equation] Approximating the sum by an integral, we get [equation] whose solution is $-z_0$ in the strong-coupling limit $λ>>1$, where $z=z_0/√(λ)$ is the saddle-point in [ref] [equation] Since the anti-symmetric representation has the symmetry $f↦1-f,z↦-z$, our result agrees completely with [ref] [equation]

Comments on Higher Rank Wilson Loops in ${\cal N}=2^*$ · Anti-symmetric representation · 0.93
This agrees with the localization prediction [ref] , which can be obtained by evaluating ([ref]) at strong coupling with $k/N$ fixed (note that, as already discussed above, the result in ([ref]) applies to the more general 1/8-BPS Wilson loops, which are just related to the 1/2-BPS one by a rescaling of the coupling $λ→λ(1-a^2)$).

Giant Wilson Loops and AdS$_2$/dCFT$_1$ · D3-brane solution in $AdS_5×S^5$ · 0.93
When evaluating the solution [ref] in the classical action [ref], the result is [equation] In the second line we have written the gravity answer using field theory variables to illustrate that it matches perfectly with the leading result for the expectation value of the $k$-antisymmetric Wilson loop computed using the matrix model [ref] .

One-loop Effective Action of the Holographic Antisymmetric Wilson Loop · Spectrum of operators on circular Wilson loops · 0.98

Cites it as background 16 papers

Wilson loops with $O(1)$ boxes are dual to fundamental strings [ref] , those with $O(N)$ boxes to D-branes [ref] , and those with $O(N^2)$ boxes to backreacted “bubbling” geometries [ref] .

Bubbling wormholes and matrix models · Introduction · 0.81
Indeed, the AdS/CFT dictionary has been enlarged to include D3 and D5 branes corresponding to Wilson loops in the symmetric and antisymmetric representations of $SU(N)$ for $N=4$ SYM [ref] .

Comments on Higher Rank Wilson Loops in ${\cal N}=2^*$ · Introduction · 0.92
In $𝒩=4$ SYM theory, $1/2$-BPS circular Wilson loops can be mapped by localization to a Gaussian matrix model [ref] .

Exact 1/N expansion of Wilson loop correlators in $\mathcal{N}=4$ Super-Yang-Mills theory · Review of the single loop case · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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