Holographic R\'enyi entropy for CFT with $W$ symmetry

Bin Chen; Jia-ju Zhang; Jiang Long

REVIEW

Holographic R\'enyi entropy for CFT with $W$ symmetry

Bin Chen, Jiang Long, Jia-ju Zhang

High Energy Physics - Theory · Submitted 19 Dec 2013 · 44 references

Pith's verdict: A careful, grind-it-out confirmation that the 1-loop Rényi entropy of W-symmetric CFTs matches higher spin AdS3 through order x^8, with the operator tables being the durable contribution.

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CitationBin Chen, Jia-ju Zhang, Jiang Long. "Holographic R\'enyi entropy for CFT with $W$ symmetry." arXiv:1312.5510 (2013). https://pith.science/paper/7HE4ISVN

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Worth your time if you work on higher spin holography or short-interval Rényi entropy; skippable otherwise. The real content is bookkeeping done carefully: the authors extend their twist-operator OPE machinery to CFTs with W3 and W4 symmetry and match it against the 1-loop handlebody partition function of higher spin AdS3 through order x^8 in the cross ratio. The new pieces are the spin-3 and spin-4 1-loop contributions (2.8) and (2.9) and, more usefully, the catalog of W-sector quasiprimaries with nonvanishing d_K in Appendices A and B. That table is what I would actually reach for later; the rest is a long expansion that mostly confirms what the character formulas already suggest. On the reader's worry about the classical saddle: I think it is over-weighted. In the ground state the W charges vanish, so the W fields first enter at O(c^0) — level 6 for W3 squared, level 8 for W4 squared — which is exactly where the 1-loop comparison lives. The classical part is the ordinary result and the footnote to [35] is adequate. The stress-test's explicit check of the n=2, x^6 coefficient is convincing, and the match is not an artifact of shared character formulas, because the d_K's are computed independently from conformal transformations. Soft spots, in proportion: the abstract's 'confirms' is a bit strong for one observable at one loop — 'supports' would be honest. The 2-loop and 3-loop results (3.17) and (3.18) have no gravity counterpart, which the authors say plainly; they are CFT curiosities for now. The vanishing 3-loop at n=3 is flagged as unexplained, and the speculation about m-loop vanishing for n up to m is just that. The overlap with Perlmutter [35] is disclosed in a note, which is the right handling. Nothing here is wrong. It is a grind-it-out confirmation paper, and the grind is done properly. Send it out.

Referee report

Major objections (1)

[Section 2, footnote 1] The paper assumes that the classical handlebody geometry is unchanged by the presence of higher spin fields, so that the classical Rényi entropy is still given by the Ryu-Takayanagi area formula. This assumption is acknowledged and cited to [35], but the paper's own justification is thin: it states that 'all the higher spin fields could be set to vanish without spoiling the underlying correspondence' without making explicit why the W charges vanish in the ground state and why this makes the classical part trivially the ordinary CFT result. Since the abstract claims a 'classical match' between gravity and CFT, the paper should state clearly that the classical match is trivial (both sides reduce to the ordinary CFT result) and that the nontrivial content is the 1-loop match. This is a framing issue rather than a correctness issue, but it is load-bearing for the abstract's claim.

Minor objections (8)

[Section 3.5, Eq. (3.24)] The sentence describing I^(2,3,4) is garbled: it mixes the 3-loop statement with the 1-loop statement, so the reader cannot tell which parts of I^(2,3,4) equal which parts of the earlier results. The sentence should be rewritten to state separately that I^(2,3,4)tree = I^(2)tree, I^(2,3,4)2-loop = I^(2,3)2-loop, I^(2,3,4)3-loop = I^(2)3-loop, and I^(2,3,4)1-loop = I^(2)1-loop + I^{spin-3,1-loop} + I^{spin-4,1-loop}.
[Section 3.2, Eq. (3.16)] There is a typo '1-1oop' in the text introducing Eq. (3.16); it should read '1-loop'.
[Section 2.2.2, Eq. (2.8)] The error term is written 'O(x)^9'; it should be 'O(x^9)'. The same typo appears in Eq. (3.18).
[Appendix A, Eq. (A.22)] The W4 normalization α_{W4} = α_{W4W4} = 1 is a choice. The paper should state explicitly that the product α_K d_K^2 is normalization-independent, so the match with the gravity result is unaffected by this choice.
[Section 3.3, paragraph after Eq. (A.20)] The claim that operators such as W3_{j1}W4_{j2} at level 7 have vanishing d_K is asserted without derivation. The paper should provide a check or a reference showing that these mixed operators do not contribute to the OPE of twist operators.
[Abstract] The phrase 'This confirms the higher spin gravity/CFT correspondence' is too strong for a 1-loop ground-state check. The paper should say 'provides a nontrivial check of' rather than 'confirms'.
[Section 4] The statement that 'there is weak sign that the m-loop correction with m ≥ 3 is vanishing for S_n, n = 2, 3, ..., m' is speculative. The paper should be clearer that this is a conjecture based on the n=3, 3-loop observation, not a derived result.
[Section 3.3, first bullet point] The paper states that 'There is no further contributions from the W-generated operators to the tree-level result' and attributes this to the absence of higher spin contributions to the classical action. It would help to state explicitly that this is because the W_j at level 3 has vanishing d_K, so the W fields first contribute at O(c^0).

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The paper

The paper claims

Two disjoint intervals in a W-symmetric conformal field theory carry a Rényi entropy whose gravity-side and field-theory-side computations agree exactly through order $x^8$ in the cross ratio, once massless spin-3 and spin-4 fields are tall

The reading

The paper asks whether the holographic recipe for entanglement entropy survives when the boundary theory has more symmetry than ordinary conformal invariance. It studies the Rényi mutual information of two short, widely separated intervals on the plane in a two-dimensional conformal field theory with $W_3$ and $W_4$ symmetry, the kind of theory conjectured to be dual to a higher spin theory of gravity in three-dimensional anti-de Sitter space. Both sides are expanded in the small cross ratio $x$ up to order $x^8$, which is far enough to expose the leading and subleading imprints of the spin-3 and spin-4 fields. The classical (order $c$) and one-loop (order $c^0$) pieces agree exactly, and the authors also record the two-loop and three-loop pieces on the field-theory side. If the match holds, it says the Ryu-Takayanagi area law plus a one-loop sum over massless fields of every spin is the right bookkeeping for entanglement entropy in higher spin holography, not just in pure gravity.

What carries the argument

Two machines carry the argument and are matched against each other. The gravity machine is the one-loop partition function on a handlebody quotient of AdS$_3$ by a Schottky group, written as a product over primitive conjugacy classes and over massless spins $s$; because the product factorizes by spin, the graviton, spin-3 and spin-4 contributions can be computed separately and simply added. The field-theory machine is the short-interval operator product expansion of the twist operators $σ$ and $σ̃$, $σ(z)σ̃(0) = c_n Σ_K d_K Σ_{m,r} a^K_m/m! · ā^K_r/r! · z^{-2h+h_K+m} z̄^{-2h̄+h̄_K+r} ∂^m∂̄^r Φ_K(0)$, summed over all independent quasiprimary operators $Φ_K$ of the replicated theory. The coefficients $d_K$ are fixed by the conformal transformation of each quasiprimary operator, and the whole two-interval partition function collapses into a sum of hypergeometric functions $F(h_K,h_K;2h_K;x)$. The nontrivial work is enumerating the quasiprimary operators through level eight, including those that mix the stress tensor with $W$ fields, and evaluating their $d_K$.

Extended reading notes

Core claim

The central claim is an exact agreement, through order $x^8$, between two computations of the Rényi mutual information $I_n$ of two disjoint intervals with small cross ratio. On the gravity side the classical piece is taken over unchanged from pure AdS$_3$ gravity, and the one-loop piece is read off from a product over primitive conjugacy classes of the Schottky group, $Z^{1-!loop} = 1/|1-q_γ^m|$ extended to a product over every massless spin $s$ with $m$ running from $s$ upward; the spin-3 and spin-4 contributions are additive and are given explicitly in (2.8) and (2.9). On the field-theory side the same quantity is extracted from the operator product expansion of twist operators, now including quasiprimary operators built from the $W$ fields alongside those of the Virasoro Verma module. The result is that $I^{(2,3)}_n$ equals the ordinary Virasoro result (3.16) plus the spin-3 term (2.8), and likewise for $W_4$, with the tree-level part (3.15) untouched by the extra symmetry. The authors also obtain the $1/c$ and $1/c^2$ corrections on the field-theory side, note that they vanish at $n=2$ in line with the genus-one partition function being one-loop exact, and observe that the three-loop term appears to vanish at $n=3$ without a known reason.

Load-bearing premise

The classical gravitational saddle is assumed to be unchanged by the presence of the higher spin fields, so that the leading, order-$c$ Rényi entropy is still given by the Ryu-Takayanagi area formula; the paper states this in Section 2 and points to a companion paper for a more careful justification.

Original abstract

In this paper we investigate the holographic R\'enyi entropy of two disjoint intervals on complex plane with small cross ratio $x$ for conformal field theory with $W$ symmetry in the ground state, which could be dual to a higher spin AdS$_3$ gravity. We focus on the cases of $W_3$ and $W_4$ symmetries. In order to see the nontrivial contributions from the $W$ fields, we calculate the R\'enyi entropy in the expansion of $x$ to order $x^8$ in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from $W$ fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the R\'enyi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of R\'enyi entanglement entropy, including the quantum correction, in both the AdS$_3$ gravity and the higher spin AdS$_3$ gravity.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,575 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.25 · compares with prior results 0.60 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

In this paper we investigate the holographic R\'enyi entropy of two disjoint intervals on complex plane with small cross ratio $x$ for conformal field theory with $W$ symmetry in the ground state, which could be dual to a higher spin AdS$_3$ gravity. We focus on the cases of $W_3$ and $W_4$ symmetries. In order to see the nontrivial contributions from the $W$ fields, we calculate the R\'enyi entropy in the expansion of $x$ to order $x^8$ in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from $W$ fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the R\'enyi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of R\'enyi entanglement entropy, including the quantum correction, in both the AdS$_3$ gravity and the higher spin AdS$_3$ gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we investigate the holographic R\'enyi entropy of two disjoint intervals on complex plane with small cross ratio $x$ for conformal field theory with $W$ symmetry in the ground state, which could be dual to a higher spin AdS$_3$ gravity.	contribution 0.96	0.79	0.05	hedged possibility 0.85
s1	We focus on the cases of $W_3$ and $W_4$ symmetries.	contribution 0.85	0.52	0.02	asserted 0.77
s2	In order to see the nontrivial contributions from the $W$ fields, we calculate the R\'enyi entropy in the expansion of $x$ to order $x^8$ in both the gravity and the CFT sides.	method 0.96	0.94	0.03	asserted 0.98
s3	In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields.	method 0.95	0.49	0.04	asserted 0.86
s4	In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from $W$ fields, besides the ones from Virasoro Verma module.	method 0.97	0.75	0.02	asserted 0.98
s5	In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the R\'enyi entropy.	result 0.87	0.94	0.07	asserted 1.00
s6	The classical and 1-loop results in the gravity and the CFT sides are in exact match.	result 0.99	0.92	0.39	asserted 1.00
s7	This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of R\'enyi entanglement entropy, including the quantum correction, in both the AdS$_3$ gravity and the higher spin AdS$_3$ gravity.	interpretation 1.00	0.80	0.90	asserted 0.83

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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The large-central-charge Renyi entropy formulas survive higher spin symmetry and unequal left/right central charges, and the short-interval contribution of any Virasoro primary is computed and matched to a bulk one-lo...

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51 arXiv papers cite this in 105 sentences

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Papers citing it · 55 sentences without a confident label

Uses its method or tool 4 papers

In SCFT$^n$ we use OPE of twist operators to calculate the partition function, and the Rényi mutual information turns out to be [ref] I_n=2n-1log[ ∑_K _K d_K^2 x^h_K _2F_1(h_K,h_K;2h_K;x) ], where the summation of $K$ is over all the linearly independent orthogonalized holomorphic quasiprimary operators $Φ_K$ in SCFT$^n$.

Holographic R\'enyi entropy for two-dimensional $\mathcal{N}$=(2,2) superconformal field theory · Contributions of $T$ in SCFT · 0.85
The Rényi mutual information can be calculated as [ref] I_n = 1n-1log[ ∑_K _K d_K^2 x^h_K+h̅_K _2F_1(h_K,h_K;2h_K;x) _2F_1(h̅_K,h̅_K;2h̅_K;x) ]. with $K$ being all the orthogonalized quasiprimary operators $Φ_K$ in $^n$.

Note on non-vacuum conformal family contributions to R\'enyi entropy in two-dimensional CFT · Non-chiral primary operator · 0.96
We use the OPE of the twist operators to do short interval expansion [ref] .

Holographic R\'enyi entropy for two-dimensional N=(1,1) superconformal field theory · Rényi entropy in SCFT side · 0.99

Uses its data or values 6 papers

The needed OPE coefficients $d_K$ have been calculated in [ref] d_T=n^2-1/12n^2, d_=(n^2-1)^2/288 n^4, d_=-(n^2-1)^2 (2 n^2(35 c+61)-93)/10368 n^6(70 c+29), d_=(n^2-1)^3/10368 n^6, d_TT^j_1j_2=18n^4c1s^4_j_1j_2+(n^2-1)^2144n^4, d_T^j_1j_2=n^2-196n^6c1s^4_j_1j_2+(n^2-1)^33456n^6, d_TTT^j_1j_2j_3=-18n^6c^21s^2_j_1j_2s^2_j_2j_3s^2_j_3j_1 +n^2-196n^6c ( 1s^4_j_1j_2+1s^4_j_2j_3+1s^4_j_3j_1 ) +(n^2-1)^31728n^6, with s_j_1j_2=sin(j_1-j_2)πn.

Short interval expansion of R\'enyi entropy on torus · OPE of twist operators · 0.95
Using the results of $d_K$ and the summation formulas in [ref] we get the $b_K$ we need.

Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT · OPE of twist operators · 0.81
For the vacuum module, we only need the quasiprimary operators $T_j$, $_j$, $T_j_1T_j_2$, with $0 ≤j≤n-1$, $0 ≤j_1 < j_2 ≤n-1$, the corresponding $d_K$ can be found in [ref] , and the corresponding $b_K$ can be found in [ref] , from which we may get a_T = -1/6, a_=0, a_TT = -1/30 c.

Corrections to holographic entanglement plateau · Low temperature case from method of twist operators · 0.96

Compares with it 5 papers

For the spin $s=3$ current, we obtain S^(3) one-loop_CDW = -x^6/12012-x^7/4290-7 x^8/16830-28 x^9/46189-15 x^10/19019-2 x^11/2093+O(x^12), and the first three terms are in agreement with the $n→1$ limit of the $O(x^8)$ calculation (2.8) in [ref] .

On the next-to-leading holographic entanglement entropy in $AdS_{3}/CFT_{2}$ · Comparison with known results · 1.00
In the special case of $d=2$, we may use $C_T = c/2π^2$ to rewrite Eq. inf in terms of the central charge $c$ of the Virasoro algebra and find at linear order in $ṇ$ intd I_n(A_1, A_2) = cṇ3 [equation] , which agrees with the result of a CFT derivation in [ref] and short interval expansions computed in [ref] .

The Gravity Dual of Renyi Entropy · Introduction · 0.99
Such orders are lower than the previous results in literature, for examples, order $x^8$ for complex plane case in [ref] and order $q^4$ for torus case in [ref] .

Holographic R\'enyi entropy for two-dimensional $\mathcal{N}$=(2,2) superconformal field theory · Conclusion and discussion · 1.00

Cites it as support 1 paper

Nevertheless, considering the accumulating evidence on the correctness of holographic computation of two-interval Rényi entropy studied in [ref] and the single-interval Rényi entropy at finite temperature case discussed above, we are inclined to believe that holographic computation of Rényi entropy in other cases, including the high temperature case, provides a trustable way to compute exactly the same entropy in 2D CFT in the large central charge limit.

Single Interval R\'enyi Entropy At Low Temperature · Conclusion and discussion · 0.99

Cites it as background 26 papers

The quasiprimary operators from vacuum module has been studied carefully in [ref] .

Holographic R\'enyi entropy in AdS$_3$/LCFT$_2$ correspondence · LCFT dual to critical CTMG · 0.90
In CFT side one can use the OPE (operator product expansion) of twist operators, and various cases have been considered [ref] .

On one-loop entanglement entropy of two short intervals from OPE of twist operators · Introduction · 0.80
Various conditions have been investigated for the holographic entropy of two short intervals on a line with zero temperature [ref] .

Holographic R\'enyi entropy for two-dimensional N=(1,1) superconformal field theory · Introduction · 0.98

What it cites · 51 sentences without a confident label

Builds on 3 papers

In this paper we continue the work initiated in [ref] .

On short interval expansion of R\'enyi entropy · Introduction · 1.00
In this paper we continue the study of the short interval expansion of Rényi entropy of two-dimensional (2D) conformal field theory (CFT) initiated in [ref] .

Entanglement Renyi entropies in holographic theories · Introduction · 0.87
In this paper we continue the study of the short interval expansion of Rényi entropy of two-dimensional (2D) conformal field theory (CFT) initiated in [ref] .

Entanglement entropy of two disjoint intervals in conformal field theory II · Introduction · 0.87

Uses the method or tool of 5 papers

The method of calculating the 1-loop correction of the Rényi entropy was given in [ref] , and it is directly related to the computation of the partition function [ref] .

Holographic entanglement beyond classical gravity · The 1-loop correction · 0.97
The method of calculating the 1-loop correction of the Rényi entropy was given in [ref] , and it is directly related to the computation of the partition function [ref] .

Partition Functions of Three-Dimensional Pure Gravity · The 1-loop correction · 0.97
The method of calculating the 1-loop correction of the Rényi entropy was given in [ref] , and it is directly related to the computation of the partition function [ref] .

One-loop Partition Functions of 3D Gravity · The 1-loop correction · 0.97

Uses data or values from 1 paper

When the spacetime is the quotient of global AdS$_3$ by a Schottky group $$, the 1-loop partition function for the graviton is [ref] Z^1-loop=∏_∈∏_m=2^inf1|1-q_^m|, with $$ being a set of representatives of the primitive conjugacy classes of $$.

One-loop Partition Functions of 3D Gravity · Graviton · 0.86

Compares with 7 papers

I_n^(2)1-loop=.+59604098747 n^2+161961045427) +O(x^9) which matches the result in [ref] to order $x^8$.

Holographic entanglement beyond classical gravity · Ordinary CFT · 0.99
Using this method, the small cross ratio $x$ expansion of the 1-loop correction of Rényi mutual information to order $x^8$ has been given in [ref] , and it has been confirmed to order $x^6$ from CFT computation by considering only the contributions from the Virasoro Verma module [ref] .

On short interval expansion of R\'enyi entropy · Graviton · 0.81
This is just ([ref]) and matches the result in [ref] .

Entanglement Entropy at Large Central Charge · Ordinary CFT · 1.00

Cites as background 26 papers

For the pure AdS gravity, it was conjectured by Ryu and Takayanagi that the leading contribution of entanglement entropy is captured by the area of a minimal surface in the bulk with asymptotic boundary ending on $A$ [ref] .

Aspects of Holographic Entanglement Entropy · Introduction · 0.95
For the pure AdS gravity, it was conjectured by Ryu and Takayanagi that the leading contribution of entanglement entropy is captured by the area of a minimal surface in the bulk with asymptotic boundary ending on $A$ [ref] .

Holographic Entanglement Entropy: An Overview · Introduction · 0.95
For the pure AdS gravity, it was conjectured by Ryu and Takayanagi that the leading contribution of entanglement entropy is captured by the area of a minimal surface in the bulk with asymptotic boundary ending on $A$ [ref] .

Entanglement Entropy from a Holographic Viewpoint · Introduction · 0.95

Discussion

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