$C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories

A.A. Tseytlin; M. Beccaria

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$C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories

M. Beccaria, A.A. Tseytlin

High Energy Physics - Theory · Submitted 30 Apr 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient $c_3$ for the higher derivative (1,0) superconformal 6d vector multiplet.

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CitationA.A. Tseytlin, M. Beccaria. "$C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories." arXiv:1705.00305 (2017). https://pith.science/paper/WT7OSMHY

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In arXiv:1510.02685 we proposed linear relations between the Weyl anomaly $c_1, c_2, c_3$ coefficients and the 4 coefficients in the chiral anomaly polynomial for (1,0) superconformal 6d theories. These relations were determined up to one free parameter $\xi$ and its value was then conjectured using some additional assumptions. A different value for $\xi$ was recently suggested in arXiv:1702.03518 using a different method. Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient $c_3$ for the higher derivative (1,0) superconformal 6d vector multiplet. This multiplet contains the 4-derivative conformal gauge vector, 3-derivative fermion and 2-derivative scalar. We find the corresponding value of $c_3$ which is proportional to the coefficient $C_T$ in the 2-point function of stress tensor using its relation to the first derivative of the Renyi entropy or the second derivative of the free energy on the product of thermal circle and 5d hyperbolic space. We present some general results of computation of the Renyi entropy and $C_T$ from the partition function on $S^1 \times \mathbb H^{d-1}$ for higher derivative conformal scalars, spinors and vectors in even dimensions. We also give an independent derivation of the conformal anomaly coefficients of the higher derivative vector multiplet from the Seeley-DeWitt coefficients on an Einstein background.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In arXiv:1510.02685 we proposed linear relations between the Weyl anomaly $c_1, c_2, c_3$ coefficients and the 4 coefficients in the chiral anomaly polynomial for (1,0) superconformal 6d theories.	background 0.99	0.64	0.06	asserted 1.00
s1	These relations were determined up to one free parameter $\xi$ and its value was then conjectured using some additional assumptions.	background 0.71	0.45	0.04	asserted 0.86
s2	A different value for $\xi$ was recently suggested in arXiv:1702.03518 using a different method.	background 0.97	0.04	0.03	asserted 0.69
s3	Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient $c_3$ for the higher derivative (1,0) superconformal 6d vector multiplet.	contribution 0.65	0.87	0.89	asserted 1.00
s4	This multiplet contains the 4-derivative conformal gauge vector, 3-derivative fermion and 2-derivative scalar.	background 0.48	0.17	0.03	asserted 1.00
s5	We find the corresponding value of $c_3$ which is proportional to the coefficient $C_T$ in the 2-point function of stress tensor using its relation to the first derivative of the Renyi entropy or the second derivative of the free energy on the product of thermal circle and 5d hyperbolic space.	result 0.62	0.94	0.19	asserted 1.00
s6	We present some general results of computation of the Renyi entropy and $C_T$ from the partition function on $S^1 \times \mathbb H^{d-1}$ for higher derivative conformal scalars, spinors and vectors in even dimensions.	contribution 0.94	0.92	0.06	asserted 1.00
s7	We also give an independent derivation of the conformal anomaly coefficients of the higher derivative vector multiplet from the Seeley-DeWitt coefficients on an Einstein background.	contribution 0.75	0.95	0.06	asserted 1.00

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In arXiv:1510.02685 we proposed linear relations between the Weyl anomaly $c_1, c_2, c_3$ coefficients and the 4 coefficients in the chiral anomaly polynomial for (1,0) superconformal 6d theories. These relations were determined up to one free parameter $\xi$ and its value was then conjectured using some additional assumptions. A different value for $\xi$ was recently suggested in arXiv:1702.03518 using a different method. Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient $c_3$ for the higher derivative (1,0) superconformal 6d vector multiplet. This multiplet contains the 4-derivative conformal gauge vector, 3-derivative fermion and 2-derivative scalar. We find the corresponding value of $c_3$ which is proportional to the coefficient $C_T$ in the 2-point function of stress tensor using its relation to the first derivative of the Renyi entropy or the second derivative of the free energy on the product of thermal circle and 5d hyperbolic space. We present some general results of computation of the Renyi entropy and $C_T$ from the partition function on $S^1 \times \mathbb H^{d-1}$ for higher derivative conformal scalars, spinors and vectors in even dimensions. We also give an independent derivation of the conformal anomaly coefficients of the higher derivative vector multiplet from the Seeley-DeWitt coefficients on an Einstein background.

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Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories
hep-th 2020-08 accept novelty 7.0 of 10

Six-dimensional superconformal field theories with a tensor branch have positive a-type Weyl anomaly, proved by showing that the mixed gauge-global anomalies that would create 2-group symmetries must vanish in SCFTs.
A Conformal Collider for Holographic CFTs
hep-th 2018-05 conditional novelty 6.0 of 10

A new collider-style argument shows that in holographic conformal field theories, the stress tensor's three-point couplings to low-spin operators are fixed by two-point functions or suppressed by the higher-spin gap.
M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture
hep-th 2018-05 accept novelty 6.0 of 10

The R4 coefficient in the 11d M-theory four-graviton amplitude is derived from 6d CFT data via the W_N chiral algebra, matching the independently known string theory value.
Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions)
hep-th 2017-05 conditional novelty 6.0 of 10

Numerical bootstrap bounds on the flavor central charge of six-dimensional theories with E8 symmetry come within a few percent of the rank-one E-string value of 150, motivating a conjecture that the E-string saturates...
Semi-universality of conformal higher-derivative and conformal higher-spin fields
hep-th 2026-06 unverdicted novelty 5.0 of 10

Thermal partition functions of conformal higher-derivative and higher-spin fields develop universal poles in (1-|ω_i|) in the semi-universal limit, with residues sensitive to negative-twist states, verified via mode s...

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AdS$_7$/CFT$_6$ with orientifolds
hep-th 2017-12 conditional novelty 5.0 of 10

The supergravity and field theory computations of the a conformal anomaly agree at leading order for six-dimensional linear quivers containing O6 and O8 orientifold planes.
Universal RG Flows Across Dimensions and Holography
hep-th 2017-08 accept novelty 5.0 of 10

Twisting a superconformal field theory along its exact R-symmetry on a compact manifold produces renormalization group flows across dimensions whose anomaly coefficients and free energies are tied by universal, theory...

How the literature uses it

20 arXiv papers cite this in 69 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 · 40 sentences without a confident label

Contests it 1 paper

In fact, the original value obtained in [ref] turned out to be incompatible with the later results of [ref] , which indicated a value $ξ=-8/9$.

Conformal anomalies for (maximal) 6d conformal supergravity · Introduction and summary · contest 0.80

Builds on it 1 paper

We show this without making any additional assumption, thus generalising the result of [ref] and providing an explanation for its extension in [ref] ; this parallels and confirms the analysis of [ref] .

Conformal anomalies for (maximal) 6d conformal supergravity · Introduction and summary · 0.95

Uses its method or tool 2 papers

We use the method of factorization [ref] to write the higher derivative operator in terms of the product of two-derivative operators.

Partition functions of higher derivative conformal fields on conformally related spaces · Introduction · 1.00
[equation] We can now apply our shortcut route by restricting to an Einstein metric without spoiling the independence of the four Weyl-invariant terms in the 6D Weyl anomaly basis [This shortcut route to the Weyl anomaly has already been noticed in [ref] and successfully put into use by [ref] .].

One-loop divergences in 7D Einstein and 6D Conformal Gravities · 6D Conformal Gravity: UV divergence · 0.86

Uses its data or values 1 paper

In [ref] , the authors computed the $C_T$ for a three-derivative Weyl fermion by studying the partition function on S$^1×ℍ^5$, and found C_T = - 72 5 (three-derivative Weyl fermion).

Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions) · Central charge $C_T$ · 0.98

Compares with it 6 papers

Following that we obtain free energy and the universal part of the Rényi entropy [equation] The Rényi entropy precisely agrees with [ref] but free energy differs by a term linear in $q$.

Partition functions of higher derivative conformal fields on conformally related spaces · Kaluza-Klein zero modes: · 0.90
In conclusion the conformal anomaly coefficients derived from [ref] with [ref] are [equation] Our result [ref] coincides with that of [ref] obtained via indirect methods,[Our values [ref] are double those given in (6.3) of [ref] , since formal Majorana-Weyl spinors are used there] thereby providing an independent direct confirmation.

Conformal anomalies in 6d 4-derivative theories: a heat-kernel analysis · Three-derivative fermion · 1.00
For $q=2$, our result correctly reproduces the free energy obtained in [ref] (with an appropriate dimension of the spinor).

Free energy and defect $C$-theorem in free fermion · Even p · 0.98

Cites it as background 7 papers

In this section we review the partition function of the free Maxwell field on $AdS_3×S^1_q$ [ref] .

Partition functions of higher derivative conformal fields on conformally related spaces · Maxwell field on $AdS_3×S^1_q$ · 0.88
A four-derivative Weyl-covariant differential operator [ref] in $d$-dimensions was constructed by Paneitz (cf. [ref] ; the $4d$ case was first given in [ref] and [ref] ), [equation] where $S_mn$ is the Schouten tensor [equation] Such operator allows one to consider the following Weyl-invariant action in $6d$ for a real scalar, from which we can compute the corresponding effective action and conformal anomaly via [ref], [equation] The operator [ref] is written in the symmetric form [ref].

Conformal anomalies in 6d 4-derivative theories: a heat-kernel analysis · Four-derivative scalar field · 0.83
In the introduction, Sections [ref] and [ref] as well as appendices [ref], [ref] and [ref], which are devoted to the evaluation of the anomaly, we adhere to the QFT notation of e.g. [ref] .

Conformal anomalies for (maximal) 6d conformal supergravity · Notation and conventions · 0.90

What it cites · 14 sentences without a confident label

Uses the method or tool of 2 papers

obtaining it from the metric variation of a Weyl-invariant action in curved background and then using 1.14.Some simplification is that in computing $⟨T T⟩$ in 1.14 one can drop total derivative (descendant) parts in one of the two $T$-factors [ref] .

$C_T$ for Non-unitary CFTs in Higher Dimensions · Introduction · 0.94
An alternative approach that we shall follow below is to exploit the relation between $C_T$ and the entropy [ref] .

A universal feature of CFT Renyi entropy · Introduction · 0.95

Uses data or values from 5 papers

In fact, as $C_T$ for the scalar and the 4-derivative vector is already known [ref] , it remains only to compute it for the 3-derivative spinor field.[We thank Ying-Hsuan Lin and Chi-Ming Chang for this suggestion.]

On $C_{J}$ and $C_{T}$ in Conformal QED · Introduction · 0.84
In fact, as $C_T$ for the scalar and the 4-derivative vector is already known [ref] , it remains only to compute it for the 3-derivative spinor field.[We thank Ying-Hsuan Lin and Chi-Ming Chang for this suggestion.]

$C_T$ for Non-unitary CFTs in Higher Dimensions · Introduction · 0.84
The 4 coefficients in 1.5 could have been be fixed also from the $å$-anomalies for the 4 multiplets: free tensor, free hyper, the (2,0) multiplet at large $N$ and the higher derivative vector multiplet (the $å$-anomaly of which was found in [ref] after [ref] already appeared).

Conformal anomaly c-coefficients of superconformal 6d theories · Introduction · 0.88

Cites as background 18 papers

This multiplet is the higher-derivative (non-unitary) superconformal 6d $(1,0)$ vector multiplet $V^(1,0)$ that contains the 4-derivative gauge vector $V^(4)_μ$ (with action $∼∫F_ ^2 F^$), the 3-derivative MW spinors $ψ^(3)$ and the 2-derivative scalars $$ [ref] (cf. 1.6) 1.10 V^(1,0) = 3φ+2ψ^(3), ++V^(4) .

Renormalizable supersymmetric gauge theory in six dimensions · Introduction · 0.85
This multiplet is the higher-derivative (non-unitary) superconformal 6d $(1,0)$ vector multiplet $V^(1,0)$ that contains the 4-derivative gauge vector $V^(4)_μ$ (with action $∼∫F_ ^2 F^$), the 3-derivative MW spinors $ψ^(3)$ and the 2-derivative scalars $$ [ref] (cf. 1.6) 1.10 V^(1,0) = 3φ+2ψ^(3), ++V^(4) .

Conformal a-anomaly of some non-unitary 6d superconformal theories · Introduction · 0.85
This multiplet is the higher-derivative (non-unitary) superconformal 6d $(1,0)$ vector multiplet $V^(1,0)$ that contains the 4-derivative gauge vector $V^(4)_μ$ (with action $∼∫F_ ^2 F^$), the 3-derivative MW spinors $ψ^(3)$ and the 2-derivative scalars $$ [ref] (cf. 1.6) 1.10 V^(1,0) = 3φ+2ψ^(3), ++V^(4) .

Conformal anomaly c-coefficients of superconformal 6d theories · Introduction · 0.85

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