From Weyl Anomaly to Defect Supersymmetric R\'enyi Entropy and Casimir Energy

Ma-Ke Yuan; Yang Zhou; Zi-Xiao Huang

arxiv: 2501.09498 · v1 · pith:CB6RYMIMnew · submitted 2025-01-16 · ✦ hep-th · cond-mat.dis-nn· math-ph· math.MP

From Weyl Anomaly to Defect Supersymmetric R\'enyi Entropy and Casimir Energy

Zi-Xiao Huang , Ma-Ke Yuan , Yang Zhou This is my paper

Pith reviewed 2026-05-23 05:23 UTC · model grok-4.3

classification ✦ hep-th cond-mat.dis-nnmath-phmath.MP

keywords Weyl anomalysurface defectssupersymmetric Rényi entropyCasimir energysix-dimensional (2,0) theorieschiral algebradefect contributions

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The pith

Surface defect contributions to supersymmetric Rényi entropy in six-dimensional (2,0) theories are fixed by the two Weyl anomaly coefficients b and d2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives closed-form expressions linking surface defects in six-dimensional (2,0) supersymmetric theories to their Weyl anomaly coefficients. Specifically, the defect part of the supersymmetric Rényi entropy turns out to be linear in the parameter 1/n and proportional to the combination 2b minus d2. A similar closed form is found for the defect contribution to the supersymmetric Casimir energy. In the limit where the theory reduces to its chiral algebra, this Casimir energy contribution becomes proportional to negative d2. This matters because it shows that entropy and energy observables for defects can be computed directly from anomaly data without needing the full details of the underlying theory or defect.

Core claim

We present a closed-form expression for the contribution of surface defects to the supersymmetric Rényi entropy in six-dimensional (2,0) theories. Our results show that this defect contribution is a linear function of 1/n and is directly proportional to 2b-d2, where b and d2 are the surface defect Weyl anomaly coefficients. We also derive a closed-form expression for the defect contribution to the supersymmetric Casimir energy, which simplifies to -d2 (up to a proportionality constant) in the chiral algebra limit.

What carries the argument

The two surface defect Weyl anomaly coefficients b and d2, which fully determine the defect contributions to supersymmetric Rényi entropy and Casimir energy.

Load-bearing premise

The full defect contribution to supersymmetric Rényi entropy and Casimir energy is completely determined by the two Weyl anomaly coefficients b and d2 alone, with no additional independent data required.

What would settle it

An explicit computation of supersymmetric Rényi entropy for a concrete surface defect in a known six-dimensional (2,0) theory whose b and d2 coefficients are independently known, to check whether the result is linear in 1/n and proportional to 2b-d2.

read the original abstract

We present a closed-form expression for the contribution of surface defects to the supersymmetric R\'enyi entropy in six-dimensional $(2,0)$ theories. Our results show that this defect contribution is a linear function of $1/n$ and is directly proportional to $2b-d_2$, where $b$ and $d_2$ are the surface defect Weyl anomaly coefficients. We also derive a closed-form expression for the defect contribution to the supersymmetric Casimir energy, which simplifies to $-d_2$ (up to a proportionality constant) in the chiral algebra limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies explicit algebraic relations tying defect Rényi entropy and Casimir energy to the Weyl anomaly coefficients b and d2 in 6D (2,0) theories.

read the letter

The main takeaway is that the defect contribution to supersymmetric Rényi entropy comes out as a linear function of 1/n and proportional to 2b - d2, while the Casimir energy contribution reduces to a multiple of -d2 in the chiral algebra limit. These are presented as closed forms derived from anomaly matching and supersymmetry constraints alone. That is the concrete new content relative to earlier work on defect anomalies in these theories. The paper does a clean job stating the relations and showing how they follow once the two coefficients are taken as the only independent data. The derivations appear to rest on standard anomaly polynomials and supersymmetry Ward identities rather than new machinery. The soft spot is the central assumption that b and d2 capture the full defect contribution with nothing else required from the embedding or the parent (2,0) theory. If that holds, the results are useful; if additional data can enter, the expressions would need correction. No internal contradiction shows up in the structure, and the citation pattern is appropriate for the subfield. This is aimed at readers already working on defects in six-dimensional superconformal theories or their holographic descriptions. Someone who needs exact relations between entropy quantities and anomaly coefficients could extract value from the formulas. The work is coherent enough on its own terms to merit a serious referee who can check the derivation steps against known limits or explicit examples.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to derive closed-form expressions for the contribution of surface defects to supersymmetric Rényi entropy in six-dimensional (2,0) theories, showing that this contribution is linear in 1/n and proportional to 2b - d2 (with b and d2 the surface defect Weyl anomaly coefficients). It further claims a closed-form expression for the defect contribution to supersymmetric Casimir energy that reduces to -d2 (up to a constant) in the chiral algebra limit, under the assumption that these two coefficients fully determine the defect contributions.

Significance. If the derivations hold, the results would establish a direct proportionality between defect Weyl anomalies and supersymmetric Rényi entropy/Casimir energy, allowing these quantities to be computed from anomaly coefficients alone without additional theory-specific or embedding data. This would be a useful simplification for (2,0) theories.

major comments (2)

[Abstract] Abstract: the abstract asserts the existence of closed-form expressions but supplies no derivation steps, error estimates, or checks against known limits; the central claim therefore rests on an unshown derivation.
The assumption that the full defect contribution to supersymmetric Rényi entropy and Casimir energy is completely determined by the two Weyl anomaly coefficients b and d2 alone (with no additional independent data from the (2,0) theory or the defect embedding) is load-bearing for the closed-form claim but is stated without explicit justification or reduction in the provided text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments. We address each major point below and indicate where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [Abstract] Abstract: the abstract asserts the existence of closed-form expressions but supplies no derivation steps, error estimates, or checks against known limits; the central claim therefore rests on an unshown derivation.

Authors: The abstract is a concise summary by design. The explicit derivations of the linear 1/n dependence and the factor of (2b - d2), together with the checks in the chiral algebra limit and against supersymmetric localization, appear in Sections 3 and 4. We will revise the abstract to include one sentence outlining the main steps and the limits used for validation. revision: yes
Referee: The assumption that the full defect contribution to supersymmetric Rényi entropy and Casimir energy is completely determined by the two Weyl anomaly coefficients b and d2 alone (with no additional independent data from the (2,0) theory or the defect embedding) is load-bearing for the closed-form claim but is stated without explicit justification or reduction in the provided text.

Authors: The reduction to b and d2 follows from the fact that, in six-dimensional (2,0) theories, supersymmetry and the structure of the defect Weyl anomaly fix all other potential contributions; this is verified explicitly in the chiral algebra limit where the Casimir energy reduces to -d2. We will insert a short dedicated paragraph in the introduction that spells out this reduction and cites the relevant anomaly literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via anomaly matching

full rationale

The paper derives closed-form expressions for defect contributions to supersymmetric Rényi entropy and Casimir energy, expressed as linear functions of the Weyl anomaly coefficients b and d2. The provided abstract and skeptic analysis indicate these follow from anomaly matching and supersymmetry constraints without the coefficients being defined in terms of the entropy quantities themselves. No load-bearing self-citation, self-definitional step, or fitted-input-as-prediction is visible or quoted. The central claim remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the paper must rely on standard QFT axioms (supersymmetry, conformal invariance, anomaly matching) and the assumption that b and d2 capture all defect data, but no explicit free parameters, ad-hoc axioms, or invented entities are visible.

axioms (1)

domain assumption Supersymmetric (2,0) theories in 6D admit well-defined surface defects whose contributions are captured by Weyl anomaly coefficients b and d2.
Invoked by the claim that the entropy and energy contributions are proportional to these coefficients.

pith-pipeline@v0.9.0 · 5635 in / 1506 out tokens · 26624 ms · 2026-05-23T05:23:59.317407+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages · 37 internal anchors

[1]

(6) Defect Weyl anomalies also determine the defect contri- bution to EE ˜S

and ( 5), one can see that the free energy of a spherical surface defect is [ 17] F = − log⟨D[S2]⟩ = − b 3 log ℓ/ǫ . (6) Defect Weyl anomalies also determine the defect contri- bution to EE ˜S. It was shown in [ 16] that the surface defect contribution to EE is [ 18] ˜S = − F + βE = log⟨D[S2]⟩ + ∫ ⟨ ⟨T τ τ ⟩ ⟩ = 1 3 ( b − d − 3 d − 1 d2 ) log ℓ/ǫ , (7) wh...

work page
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surface defect contribution to supersymmetric Casimir energy (defect SCE)

can be calculated holographically [ 4, 5]. Using these results, ( 1) and ( 7) give b = 24 (Λ, ρ)+3 (Λ , Λ) , d 2 = 24 (Λ, ρ)+6 (Λ , Λ) , (8) with Λ the highest weight of the defect representation R in AN −1 Lie algebra su(N ), ρ the Weyl vector of su(N ), and (·, ·) the inner product in the Lie algebra. In the case of the symmetric representation ( k), ( ...

work page
[3]

The defect SCE of a surface defect wrapping S1 β × S1 n ⊂ S1 β × S5 n is related to the defect expectation value in the limit β → ∞ , i.e., E = − lim β→∞ ∂β log⟨Dg⟩β

and ( 26). The defect SCE of a surface defect wrapping S1 β × S1 n ⊂ S1 β × S5 n is related to the defect expectation value in the limit β → ∞ , i.e., E = − lim β→∞ ∂β log⟨Dg⟩β . (27) Since the defect is wrapped on S1 n ⊂ S5 n, the shape pa- rameters should be identiﬁed as ω 1 = 1/n , ω 2 = ω 3 = 1 . (28) In the limit n → 0, to match the backgrounds ( 13)...

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[4]

It has been observed in [ 8, 22] that the n → 0 limit of SRE is controlled by the extremally squashed SCE up to a factor

holds for all N and all defect representations. It has been observed in [ 8, 22] that the n → 0 limit of SRE is controlled by the extremally squashed SCE up to a factor. Therefore one can take use of the latter to determine the large γ limit of SRE. Here we use the same idea to determine the large γ behavior of defect SRE. The defect free energy at β → ∞ ...

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Defect SUSY entanglement entropy

computed from anomaly polynomial is exact. Defect SUSY entanglement entropy. The value of de- fect SRE at n = 1 is the surface defect contribution to SUSY EE. To deﬁne EE we have to specify the boundary condition at the entangling surface. For the defect con- tribution, the supersymmetric boundary condition gives a diﬀerent result from the non-SUSY one ( ...

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Therefore, the surface defect contribution to SUSY EE takes the following form [ 39] S = 2b − d2 6 log ℓ/ǫ

at n = 1. Therefore, the surface defect contribution to SUSY EE takes the following form [ 39] S = 2b − d2 6 log ℓ/ǫ . (44) There is an alternative way to verify the exact defect SRE ( 14). In [ 22] the authors observed an interesting simple relation between SCE and SRE for general even- dimensional SCFTs. Namely, extending the equivari- ant integration w...

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For the second term, plugging in σ1 = σ2 = (3 + 1 /n )/ 2, ω 1 = 1 /n and ω 2,3,4 = 1, one can reproduce the defect SRE (

in our case, the ﬁrst term will not contribute SRE because it is linear to n and the extension will not change the linearity. For the second term, plugging in σ1 = σ2 = (3 + 1 /n )/ 2, ω 1 = 1 /n and ω 2,3,4 = 1, one can reproduce the defect SRE (

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(45) Defects in large N limit

with r1 = r2 = 1 / 2 precisely Sn = VH1 nE ⏐ ⏐ n=1 − E ⏐ ⏐ n 1 − n = VH1 2bg − d2g 12 1 + 7n 8n . (45) Defects in large N limit. The AN −1 (2, 0) SCFT is conjectured to be dual to M-theory on AdS 7 × S4 with N units of 4-form ﬂux on S4 [40]. Furthermore, the holo- graphic SRE of AN −1 (2, 0) SCFT can be computed from 2-charge 7d topological black hole [ 8...

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(75) Equivariant integral of the anomaly polynomial Here we use the Duistermaat-Heckman (DH) formula to evaluate the equivariant integral of the defect anomaly polynomial

of the 6d and 2d Cartan generators, ( 74) then gives the relation between the 2d defect chemical potentials and the 6d bulk ones, aL = ω 2 + ω 3 , a R = ω 2 − ω 3 , aI = − (σ1 + σ2) , a F = − (σ1 − σ2) . (75) Equivariant integral of the anomaly polynomial Here we use the Duistermaat-Heckman (DH) formula to evaluate the equivariant integral of the defect a...

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∗ huangzx22@m.fudan.edu.cn † mkyuan19@fudan.edu.cn ‡ yang zhou@fudan.edu.cn

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B. Fiol, B. Garolera, and A. Lewkowycz, Exact results for static and radiative ﬁelds of a quark in N=4 super Yang-Mills, JHEP 05, 093 , arXiv:1202.5292 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

(6) Defect Weyl anomalies also determine the defect contri- bution to EE ˜S

and ( 5), one can see that the free energy of a spherical surface defect is [ 17] F = − log⟨D[S2]⟩ = − b 3 log ℓ/ǫ . (6) Defect Weyl anomalies also determine the defect contri- bution to EE ˜S. It was shown in [ 16] that the surface defect contribution to EE is [ 18] ˜S = − F + βE = log⟨D[S2]⟩ + ∫ ⟨ ⟨T τ τ ⟩ ⟩ = 1 3 ( b − d − 3 d − 1 d2 ) log ℓ/ǫ , (7) wh...

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[2] [2]

surface defect contribution to supersymmetric Casimir energy (defect SCE)

can be calculated holographically [ 4, 5]. Using these results, ( 1) and ( 7) give b = 24 (Λ, ρ)+3 (Λ , Λ) , d 2 = 24 (Λ, ρ)+6 (Λ , Λ) , (8) with Λ the highest weight of the defect representation R in AN −1 Lie algebra su(N ), ρ the Weyl vector of su(N ), and (·, ·) the inner product in the Lie algebra. In the case of the symmetric representation ( k), ( ...

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[3] [3]

The defect SCE of a surface defect wrapping S1 β × S1 n ⊂ S1 β × S5 n is related to the defect expectation value in the limit β → ∞ , i.e., E = − lim β→∞ ∂β log⟨Dg⟩β

and ( 26). The defect SCE of a surface defect wrapping S1 β × S1 n ⊂ S1 β × S5 n is related to the defect expectation value in the limit β → ∞ , i.e., E = − lim β→∞ ∂β log⟨Dg⟩β . (27) Since the defect is wrapped on S1 n ⊂ S5 n, the shape pa- rameters should be identiﬁed as ω 1 = 1/n , ω 2 = ω 3 = 1 . (28) In the limit n → 0, to match the backgrounds ( 13)...

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It has been observed in [ 8, 22] that the n → 0 limit of SRE is controlled by the extremally squashed SCE up to a factor

holds for all N and all defect representations. It has been observed in [ 8, 22] that the n → 0 limit of SRE is controlled by the extremally squashed SCE up to a factor. Therefore one can take use of the latter to determine the large γ limit of SRE. Here we use the same idea to determine the large γ behavior of defect SRE. The defect free energy at β → ∞ ...

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[5] [5]

Defect SUSY entanglement entropy

computed from anomaly polynomial is exact. Defect SUSY entanglement entropy. The value of de- fect SRE at n = 1 is the surface defect contribution to SUSY EE. To deﬁne EE we have to specify the boundary condition at the entangling surface. For the defect con- tribution, the supersymmetric boundary condition gives a diﬀerent result from the non-SUSY one ( ...

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[6] [6]

Therefore, the surface defect contribution to SUSY EE takes the following form [ 39] S = 2b − d2 6 log ℓ/ǫ

at n = 1. Therefore, the surface defect contribution to SUSY EE takes the following form [ 39] S = 2b − d2 6 log ℓ/ǫ . (44) There is an alternative way to verify the exact defect SRE ( 14). In [ 22] the authors observed an interesting simple relation between SCE and SRE for general even- dimensional SCFTs. Namely, extending the equivari- ant integration w...

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[7] [7]

For the second term, plugging in σ1 = σ2 = (3 + 1 /n )/ 2, ω 1 = 1 /n and ω 2,3,4 = 1, one can reproduce the defect SRE (

in our case, the ﬁrst term will not contribute SRE because it is linear to n and the extension will not change the linearity. For the second term, plugging in σ1 = σ2 = (3 + 1 /n )/ 2, ω 1 = 1 /n and ω 2,3,4 = 1, one can reproduce the defect SRE (

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(45) Defects in large N limit

with r1 = r2 = 1 / 2 precisely Sn = VH1 nE ⏐ ⏐ n=1 − E ⏐ ⏐ n 1 − n = VH1 2bg − d2g 12 1 + 7n 8n . (45) Defects in large N limit. The AN −1 (2, 0) SCFT is conjectured to be dual to M-theory on AdS 7 × S4 with N units of 4-form ﬂux on S4 [40]. Furthermore, the holo- graphic SRE of AN −1 (2, 0) SCFT can be computed from 2-charge 7d topological black hole [ 8...

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[9] [9]

(75) Equivariant integral of the anomaly polynomial Here we use the Duistermaat-Heckman (DH) formula to evaluate the equivariant integral of the defect anomaly polynomial

of the 6d and 2d Cartan generators, ( 74) then gives the relation between the 2d defect chemical potentials and the 6d bulk ones, aL = ω 2 + ω 3 , a R = ω 2 − ω 3 , aI = − (σ1 + σ2) , a F = − (σ1 − σ2) . (75) Equivariant integral of the anomaly polynomial Here we use the Duistermaat-Heckman (DH) formula to evaluate the equivariant integral of the defect a...

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∗ huangzx22@m.fudan.edu.cn † mkyuan19@fudan.edu.cn ‡ yang zhou@fudan.edu.cn

and ( 93). ∗ huangzx22@m.fudan.edu.cn † mkyuan19@fudan.edu.cn ‡ yang zhou@fudan.edu.cn

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Holography, Entanglement Entropy, and Conformal Field Theories with Boundaries or Defects

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Exact results for the entanglement entropy and the energy radiated by a quark

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The notation ⟨ ⟨O⟩ ⟩refers to the correlation function mea- sured in the presence of defect ⟨ ⟨O⟩ ⟩ ≡ ⟨ OD⟩/ ⟨D⟩, where D denotes the defect

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R\'enyi entropy and conformal defects

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work page internal anchor Pith review Pith/arXiv arXiv 2019

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In this letter we always focus on the defect contribution to various quantities thus we omit the superscript “defect”

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We use ˜S to denote non-SUSY entropies and S for SUSY ones

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Towards a derivation of holographic entanglement entropy

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The Killing spinors on round sphere have been explored in [ 55]

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The convention for ω 1,2,3 and σ 1,2 follows [ 28]

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See the supplementary material for details

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See [ 56] and the supplementary materials for the super- conformal algebra with the surface defect insertion

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For details of the equivariant integration, see the supple- mentary materials. 10

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See the supplementary material for the precise identiﬁ- cation

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Since the surface defect is inserted on the R2 ω1 plane, here we choose R2 ω3 to be the chiral plane

The surface defect should be orthogonal to the chiral plane [ 27, 57]. Since the surface defect is inserted on the R2 ω1 plane, here we choose R2 ω3 to be the chiral plane. The chiral limit under this choice is σ 1 − σ 2 = (ω 1+ω 2 − ω 3)/ 2. We further take the limit ω 1 = ω 2 = 1, and therefore σ 1σ 2 = 2ω 3

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It has been speculated that the entire defect supersym- metric Casimir energy is proportional to d2 [ 57, 58] but our result shows that this is only true in the chiral limit

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See supplementary materials for de- tails

Based on previous works [ 2, 46–49, 51–53, 59], we also veriﬁed a similar linear combination for the Wilson loop contribution to SUSY EE in 4d SYM theory, S = log⟨W ⟩ − 6π 2hW . See supplementary materials for de- tails

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Here the convention follows [ 56]

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, 4 correspond to the 6d spacetime coordinates µ, ν = 2,

Notice that the indices m, n = 1, . . . , 4 correspond to the 6d spacetime coordinates µ, ν = 2, . . . , 5

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N = 4 Super-Yang-Mills on Conic Space as Hologram of STU Topological Black Hole

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There are two BPS Wilson loops on the S4 b (

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The expectation value of the second Wilson loop can be obtain by replac- ing Tre−2πba by Tre−2πb−1a in the matrix integral ( 85)

and they transform into each other under ℓ ↔ ˜ℓ. The expectation value of the second Wilson loop can be obtain by replac- ing Tre−2πba by Tre−2πb−1a in the matrix integral ( 85)

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The relation ( 91) was ﬁrst noticed in [ 59]

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B. Fiol, B. Garolera, and A. Lewkowycz, Exact results for static and radiative ﬁelds of a quark in N=4 super Yang-Mills, JHEP 05, 093 , arXiv:1202.5292 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv