REVIEW
Conformal anomaly of (2,0) tensor multiplet in six dimensions and AdS/CFT correspondence
High Energy Physics - Theory · Submitted 9 Jan 2000 · 56 references
Pith's verdict: A clean, explicit heat-kernel computation that settles the d=6 tensor-multiplet anomaly in the conformal basis and makes the type-B match / type-A mismatch with AdS7 supergravity a precise statement.
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Referee report
The paper
The paper claims
The free six-dimensional (2,0) tensor multiplet's Weyl anomaly matches the large-N M5-brane prediction in its Weyl-invariant part, but not in its Euler-density part.
The reading
What carries the argument
The computation reduces to the Seeley-DeWitt coefficient $b_6$ of the relevant second-order Laplace operators: the conformal scalar, the squared Dirac operator, and the Hodge-DeRham operator on two-forms, with ghost subtractions. The raw heat-kernel invariants are then rewritten in the conformal basis $E_6, I_1, I_2, I_3$ using the identities in Appendix A; this basis conversion is what makes the type A and type B coefficients comparable to the supergravity anomaly. The multiplet is assembled as half the anomaly of a non-chiral (2,2) model with ten scalars, two Dirac fermions, and one non-chiral two-form.
Extended reading notes
Core claim
The paper's central claim is that the free (2,0) tensor multiplet's conformal anomaly, when written in the standard six-dimensional basis, has a Weyl-invariant (type B) part that is exactly $4N^3$ times the corresponding supergravity prediction for $N$ coincident M5-branes. The type A part, proportional to the Euler density, does not share that factor: its coefficient is off by $4/7$ relative to $4N^3$ times the free-multiplet value, so the free multiplet anomaly is nonzero on a Ricci-flat background even though the large-N supergravity anomaly vanishes there. The authors present this as evidence that the type B coefficients are protected by the same structure that fixes two- and three-point stress-tensor correlators, while the type A coefficient probes a four-point structure and can carry interacting dynamics.
Load-bearing premise
The load-bearing premise is that the conversion of the computed Seeley-DeWitt coefficient into the $E_6$, $I_1$, $I_2$, $I_3$ basis is correct, including the definition of $I_3$ up to a total derivative, because the entire coefficient comparison depends on that step.
Editorial extensions
If this is right
- If the type B matching is exact, then the scale-anomaly coefficients of the large-N (2,0) theory are reproduced by $4N^3$ free tensor multiplets, extending the same counting that worked for two- and three-point stress-tensor correlators and graviton absorption.
- The type A coefficient is predicted to differ from the free-field value by the factor $4/7$, so any independent derivation of the Euler anomaly in the interacting theory should show that gap.
- Because the free multiplet anomaly does not vanish on a Ricci-flat background, the paper concludes that the simplest free-field model does not mimic the large-N anomaly on that background.
- The effective-action argument implies that the three Weyl-invariant anomaly coefficients are fixed by the $h^2$ and $h^3$ terms of the nonlocal effective action, i.e. by two- and three-point stress-tensor data.
- The Euler term is tied to an $h^4$ term, so a four-point stress-tensor calculation is the natural place to look for the missing type A dynamics.
Reading between the lines
- A natural next check is whether the $4/7$ discrepancy in the Euler coefficient persists beyond the free multiplet, for example in the $R^4$ correction to eleven-dimensional supergravity; if it does, the type A anomaly becomes a clean probe of interacting M5-brane dynamics.
- The same effective-action logic suggests that in any even dimension the type B coefficients are fixed by lower-point stress-tensor correlators, while the Euler coefficient requires the next-point correlator; six dimensions is the first case where that split is visible.
- If the type B matching is protected by supersymmetry, one would expect it to survive at finite $N$ only if the relevant correlators are non-renormalized; testing that would require a separate calculation.
- The paper compares only with the supergravity limit, so it does not by itself say whether the type A coefficient receives $1/N$ corrections; a finite-$N$ computation would be the decisive extension.
Original abstract
We compute the conformal anomaly of free d=6 superconformal (2,0) tensor multiplet on generic curved background. Up to a trivial covariant total-derivative term, it is given by the sum of the type A part proportional to the 6-d Euler density, and the type B part containing three independent Weyl invariants. Multiplied by factor 4N^3, the type B part of the anomaly reproduces exactly the corresponding part of the conformal anomaly of large N multiple M5-brane (2,0) theory as predicted (hep-th/9806087) by the AdS/CFT correspondence. The coefficients of the type A anomaly differ by the factor 4/7 x 4 N^3, so that the free tensor multiplet anomaly does not vanish on a Ricci-flat background. The coefficient 4N^3 is the same as found (hep-th/9703040) in the comparison of the tensor multiplet theory and the d=11 supergravity results for the absorption cross-sections of gravitons by M5 branes, and in the comparison (hep-th/9911135) of 2- and 3-point stress tensor correlators of the free tensor multiplet with the AdS_7 supergravity predictions. The reason for this coincidence is that the three Weyl-invariant terms in the anomaly are related to the $h^2$ and $h^3$ terms in the near flat space expansion of the corresponding non-local effective action, and thus to the 2-point and 3-point stress tensor correlators in flat background. At the same time, the type A anomaly is related to the $h^4$ term in the non-local part of the effective action, i.e. to a certain structure in the 4-point correlation function of stress tensors.
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
We compute the conformal anomaly of free d=6 superconformal (2,0) tensor multiplet on generic curved background. Up to a trivial covariant total-derivative term, it is given by the sum of the type A part proportional to the 6-d Euler density, and the type B part containing three independent Weyl invariants. Multiplied by factor 4N^3, the type B part of the anomaly reproduces exactly the corresponding part of the conformal anomaly of large N multiple M5-brane (2,0) theory as predicted (hep-th/9806087) by the AdS/CFT correspondence. The coefficients of the type A anomaly differ by the factor 4/7 x 4 N^3, so that the free tensor multiplet anomaly does not vanish on a Ricci-flat background. The coefficient 4N^3 is the same as found (hep-th/9703040) in the comparison of the tensor multiplet theory and the d=11 supergravity results for the absorption cross-sections of gravitons by M5 branes, and in the comparison (hep-th/9911135) of 2- and 3-point stress tensor correlators of the free tensor multiplet with the AdS_7 supergravity predictions. The reason for this coincidence is that the three Weyl-invariant terms in the anomaly are related to the $h^2$ and $h^3$ terms in the near flat space expansion of the corresponding non-local effective action, and thus to the 2-point and 3-point stress tensor correlators in flat background. At the same time, the type A anomaly is related to the $h^4$ term in the non-local part of the effective action, i.e. to a certain structure in the 4-point correlation function of stress tensors.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute the conformal anomaly of free d=6 superconformal (2,0) tensor multiplet on generic curved background. | contribution 1.00 | 0.96 | 0.09 | asserted 1.00 |
| s1 | Up to a trivial covariant total-derivative term, it is given by the sum of the type A part proportional to the 6-d Euler density, and the type B part containing three independent Weyl invariants. | result 1.00 | 0.86 | 0.20 | asserted 1.00 |
| s2 | Multiplied by factor 4N^3, the type B part of the anomaly reproduces exactly the corresponding part of the conformal anomaly of large N multiple M5-brane (2,0) theory as predicted (hep-th/9806087) by the AdS/CFT correspondence. | result 0.94 | 0.71 | 0.54 | asserted 1.00 |
| s3 | The coefficients of the type A anomaly differ by the factor 4/7 x 4 N^3, so that the free tensor multiplet anomaly does not vanish on a Ricci-flat background. | result 0.98 | 0.82 | 0.18 | asserted 1.00 |
| s4 | The coefficient 4N^3 is the same as found (hep-th/9703040) in the comparison of the tensor multiplet theory and the d=11 supergravity results for the absorption cross-sections of gravitons by M5 branes, and in the comparison (hep-th/9911135) of 2- and 3-point stress tensor correlators of the free tensor multiplet with the AdS_7 supergravity predictions. | result 0.68 | 0.13 | 0.11 | asserted 1.00 |
| s5 | The reason for this coincidence is that the three Weyl-invariant terms in the anomaly are related to the $h^2$ and $h^3$ terms in the near flat space expansion of the corresponding non-local effective action, and thus to the 2-point and 3-point stress tensor correlators in flat background. | interpretation 1.00 | 0.48 | 0.09 | asserted 1.00 |
| s6 | At the same time, the type A anomaly is related to the $h^4$ term in the non-local part of the effective action, i.e. to a certain structure in the 4-point correlation function of stress tensors. | interpretation 0.97 | 0.56 | 0.06 | asserted 1.00 |
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References
How later papers used it
56 later papers cite it
4 later papers by other authors are listed below: 1 used its method or tool and 3 used its data or values. 3 more papers by the same authors are listed: 1 used its method or tool and 2 used its data or values.
Uses its method or tool 2 of 3 papers with this label · 1 by the same authors
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Followed the conventions given in Appendix A of this paper, which also give explicit expressions for the A_a's and V_a's.
One-Loop Holographic Weyl Anomaly in Six Dimensions (2017) · 0.86 · paraphrased: the licence does not allow quoting
By the same authors
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Found the contribution of the A_mn gauge field to the one-loop partition function on S^1 x S^5 by relaxing the self-duality condition on its strength H_mnk and taking half of the resulting free energy, citing this paper and references there.
Large N expansion of superconformal index of k=1 ABJM theory and semiclassical M5 brane partition function (2023) (same authors) · 0.97 · paraphrased: the licence does not allow quoting
Uses its data or values 5 of 9 papers with this label · 2 by the same authors
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Used the ratios between the trace anomalies of one complex scalar field, one Weyl fermion and one two-form field from this paper to address the discrepancy between the entanglement entropy of a 2-form field and its trace anomaly coefficient.
Renyi Entropy of Free (2,0) Tensor Multiplet and its Supersymmetric Counterpart (2015) · 1.00 · paraphrased: the licence does not allow quoting
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Used the values from this paper to show that the logarithmic term vanishes in the interacting (2,0) conformal theory dual to supergravity on AdS_7.
Entanglement entropy of black holes (2011) · 0.92 · paraphrased: the licence does not allow quoting
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Extracted the anomaly coefficients in Table 1 from this paper, having redefined the third invariant I_3 in the structure of the anomaly functional.
The Dilaton Wess-Zumino Action in 6 Dimensions from Weyl Gauging: Local Anomalies and Trace Relations (2013) · 1.00 · paraphrased: the licence does not allow quoting
By the same authors
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Found from the expressions in this paper that for the (1,0) scalar multiplet the values are (-1/27, -1/540, 1/180).
Conformal a-anomaly of some non-unitary 6d superconformal theories (2015) (same authors) · 0.86 · paraphrased: the licence does not allow quoting
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We now list the general results for the coefficients corresponding to the operator [ref] up to $a_3(x)$, as taken from [ref] .
Six-dimensional one-loop divergences in quantum gravity from the $\mathcal{N}=4$ spinning particle (2023) (same authors) · Heat kernel and one-loop effective action · 0.97
How the literature uses it
54 arXiv papers cite this in 108 sentences
Papers citing it · 74 sentences without a confident label
Uses its method or tool 3 papers
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The relevant heat-kernel coefficient of the individual Laplacians can be worked out (see e.g. [ref] ) and the raw result on a 6D Einstein metric, modulo trivial total derivatives, reads [For notation and conventions we refer to [ref] .]
Holographic Weyl anomaly for GJMS operators: one Laplacian to rule them all · Factorization and heat kernel at 6D: four birds, one stone · 0.88
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We follow the conventions spelled out in Appendix A of [ref] , which also give explicit expressions for the $A_a$'s and $V_a$'s.
One-Loop Holographic Weyl Anomaly in Six Dimensions · Heat kernel coefficients · 0.86
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The contribution of the $A_mn$ gauge field in 3.6 to the one-loop partition function on $S^1 ×S^5$ is straightforward to find by first relaxing the self-duality condition on its strength $H_mnk$ and taking half of the resulting value of free energy at the end ( [ref] and references there).
Large N expansion of superconformal index of k=1 ABJM theory and semiclassical M5 brane partition function · Antisymmetric tensor · 0.97
Uses its data or values 8 papers
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The coefficients for $ϕ$, $ψ$, $A_μ$ and $B_μν$ were computed in [ref] , while the remaining ones are worked out in Appendix [ref].
One-Loop Holographic Weyl Anomaly in Six Dimensions · Heat kernel coefficients · 0.89
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Expressing the a-anomaly and the Casimir energy for these multiplets in terms of their values for the (2,0) tensor multiplet found in [ref] we obtain 3.5 å(S^(1,0), T^(1,0), V^(1,0)) = (11/210, 199/210, -251/210) å(T^(2,0)), E_c(S^(1,0), T^(1,0), V^(1,0)) = (37/250, 213/250, -377/250) E_c(T^(2,0)), å(T^(2,0)) = -7/1152, E_c(T^(2,0)) = -25/384 .
Conformal a-anomaly of some non-unitary 6d superconformal theories · Unitary and non-unitary low spin 6d conformal multiplets · 0.95
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Thus in the (2,0) case one should have in 41f16Note that on a Ricci flat background one gets then $b_6|_R_=0 = (-å) _6 ,$ which is the analog of the familiar relation in 4d. b_6 = - å _6 + _6 , = 196_1 . 66 For a single (2,0) tensor multiplet (denoted below as $^(2,0)$) one finds [ref] å(^(2,0)) = -71152 , 08 (^(2,0))= - 1288 .
On anomaly free 4d $\mathcal{N}$=4 and 6d (2,0) conformal supergravities and UV finiteness of Poincar\'e supergravities · 6d (2,0) conformal supergravity · 0.99
Compares with it 5 papers
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Finally, comparing with the expression ([ref]), we obtain the four anomaly coefficients in agreement with [ref] [The apparent mismatch with the original computation in [ref] is simply due to the fact that $“I_3 "$ there, actually $M_3$ in the basis of [ref] , contains a spurious contribution from the Euler density and it is not classifiable as the third type-B anomaly, as pointed out in [ref] .]
Simple recipe for holographic Weyl anomaly · $AdS_7/CFT_6$: $a, c_1, c_2$ and $c_3$ · 0.96
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A few consistency checks are in order. * The entanglement entropy ([ref]) is consistent with the result (2.12) in [ref] for a spherical entangling surface $S^4$, if one takes into account a factor difference $167$N$^3$ [ref] between the trace anomaly of $(2,0)$ tensor multiplet and that of the large-N theory of coincident M5-branes.
Renyi Entropy of Free (2,0) Tensor Multiplet and its Supersymmetric Counterpart · (2,0) tensor multiplet · 0.97
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Notice that the ratios between the $c_i$ are the same as the ones for the $(2,0)$ theory [ref] .
6d holographic anomaly match as a continuum limit · Leading behavior in the holographic limit · 0.99
Cites it as background 12 papers
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The conformal anomaly of a classically Weyl invariant theory in 6d has the following general form [ref]
Conformal a-anomaly of some non-unitary 6d superconformal theories · Introduction · 0.85
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where $𝒳_4$ is the Euler density associated to the induced metric $γ_ij$ and now [equation] Similarly, the coefficients $A$, $B_1$, $B_2$ and $B_3$ are the ones appearing in the trace anomaly, which in this case takes the form [ref] [equation] where $𝒳_6$ is the Euler density and the $I_i$ are cubic conformal invariants given by [equation]
Holographic entanglement entropy for perturbative higher-curvature gravities · Six dimensions · 0.86
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The conformal anomaly of a classically Weyl invariant theory in 6d depends on 4 independent coefficients $å,_1,_2,_3$ [ref]
$C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories · Introduction · 0.98
Discussion
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