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

TL;DR

This paper confirms the correct value of the Weyl anomaly coefficient c_3 for higher derivative (1,0) superconformal 6d vector multiplets, linking it to the C_T coefficient and refining anomaly relations.

Contribution

It provides an independent verification of the Weyl anomaly coefficient c_3 for (1,0) superconformal 6d vector multiplets, clarifying the value of the free parameter in anomaly relations.

Findings

Confirmed the value of c_3 coefficient for the vector multiplet.

Derived relations between c_3, C_T, and entropy derivatives.

Computed conformal anomalies using Seeley-DeWitt coefficients.

Abstract

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.

Tables

Table 1. Table 1: Short multiplets V p ( 1 , 0 ) subscript superscript 𝑉 1 0 𝑝 V^{(1,0)}_{p} of O ​ S ​ p ​ ( 2 , 6 | 2 ) 𝑂 𝑆 𝑝 2 conditional 6 2 OSp(2,6|2) that appear in tensor product of p 𝑝 p copies of ( 1 , 0 ) 1 0 (1,0) doubleton (hypermultiplet) representation.

Equations

$$(4\pi {)}^3⟨T_{\mu }^{\mu }⟩=-\mathrm{a}{E}_6+{\mathrm{c}}_1{I}_1+{\mathrm{c}}_2{I}_2+{\mathrm{c}}_3{I}_3,$$

$${\mathrm{c}}_3=-\frac{1}{6}({\mathrm{c}}_1-2{\mathrm{c}}_2).$$

$${\rm a}=\textstyle-{1\over 72}\big{(}\alpha-\beta+\gamma+\frac{3}{8}\,\delta\big{)}\ .$$

$$S^{(1,0)}=4\varphi+2\psi^{-},\ \ \qquad T^{(1,0)}=\varphi+2\psi^{-}+T^{-}\ ,\ \ \qquad T^{(2,0)}=S^{(1,0)}+T^{(1,0)}\$$

$${\mathrm{c}}_1$$

$${\mathrm{c}}_2$$

$${\xi }_{{}_{\mathrm{BT}}}=-\frac{31}{27}.$$

$${\xi }_{{}_{\mathrm{YZ}}}=-\frac{8}{9},$$

$$V^{(1,0)}=3\phi +2{\psi }^{(3),+}+V^{(4)}.$$

$$(\alpha ,\beta ,\gamma ,\delta )=(-1,-\frac{1}{2},-\frac{7}{240},\frac{1}{60}),$$

$${\mathrm{c}}_1=\frac{40}{189}-\frac{3}{7}\xi ,{\mathrm{c}}_2=\frac{551}{1890}+\frac{3}{28}\xi ,{\mathrm{c}}_3=\frac{13}{210}+\frac{3}{28}\xi .$$

$${\mathrm{c}}_3=\frac{5}{3\times 7!}{C}_{T,6},$$

$$⟨T^{\mu \nu }(x)T^{\rho \sigma }(0)⟩=\frac{C_{T,d}}{[{\mathrm{V}}_{S^d}{]}^2(x^2{)}^d}{\mathcal{I}}^{\mu \nu \rho \sigma }(x),$$

$${\mathrm{V}}_{S^d}=\frac{2{\pi }^{\frac{d}{2}}}{\Gamma (\frac{d}{2})},{\mathcal{I}}^{\mu \nu \rho \sigma }=\frac{1}{2}(I^{\mu \sigma }{I}^{\nu \rho }+I^{\mu \rho }{I}^{\nu \sigma })-\frac{1}{d}{\eta }^{\mu \nu }{\eta }^{\sigma \rho },I^{\mu \nu }={\eta }^{\mu \nu }-\frac{2}{x^2}{x}^{\mu }{x}^{\nu }.$$

$$C_{T,d}(\phi )=\frac{d}{d-1},C_{T,d}(\psi )=\frac{1}{2}{n}_{f}d,$$

$$C_{T,6}(V^{(4)})=-90,$$

$$C_{T,6}(V^{(1,0)})=3\times \frac{6}{5}+2\times C_{T,6}({\psi }^{(3)})-90.$$

$${\xi }_{{}_{\mathrm{BT}}}=-\frac{31}{27}\to C_{T,6}({\psi }^{(3)})=-\frac{246}{5},$$

$${\xi }_{{}_{\mathrm{YZ}}}=-\frac{8}{9}\to C_{T,6}({\psi }^{(3)})=-\frac{36}{5}.$$

$${\text{SS}}_1^{\prime }=-{\overline{\mathrm{V}}}_{{\mathbb{H}}^{d-1}}\frac{{\pi }^{\frac{d}{2}+1}\Gamma (\frac{d}{2})(d-1)}{(d+1)![{\mathrm{V}}_{S^d}{]}^2}{C}_{T,d}.$$

$${\mathrm{V}}_{{\mathbb{H}}^{d-1}}\equiv {\overline{\mathrm{V}}}_{{\mathbb{H}}^{d-1}}\log {\Lambda }_{\mathrm{IR}},{\overline{\mathrm{V}}}_{{\mathbb{H}}^{d-1}}=(-1{)}^{\frac{d}{2}-1}\frac{2{\pi }^{\frac{d}{2}-1}}{\Gamma (\frac{d}{2})}.$$

$${\overline{\mathrm{V}}}_{{\mathbb{H}}^3}=-2\pi ,{\overline{\mathrm{V}}}_{{\mathbb{H}}^5}={\pi }^2,{\overline{\mathrm{V}}}_{{\mathbb{H}}^7}=-\frac{{\pi }^3}{3},...$$

$$C_{T,4}=80{\text{SS}}_1^{\prime },C_{T,6}=-504{\text{SS}}_1^{\prime },C_{T,8}=2880{\text{SS}}_1^{\prime },...$$

$$ds^{2}_{q\rm-sphere}=\sin^{2}\theta\,q^{2}d\tau^{2}+d\theta^{2}+\cos^{2}\theta\ d\Omega^{2}_{d-2}=\sin^{2}\theta\,\big{(}q^{2}d\tau^{2}+d\rho^{2}+\sinh^{2}\rho\,d\Omega^{2}_{d-2}\big{)}=\cosh^{-2}\rho\,ds^{2}_{S^{1}_{q}\times{\mathbb{H}}^{d-1}}$$

$$I=\frac{1}{2}\int d^{d}x\sqrt{g}\Phi \mathcal{O}\Phi ,$$

$$F_q=-\log Z_q=\frac{1}{2}\log \det \mathcal{O}.$$

$$F_q\equiv {\mathcal{F}}_q\log {\Lambda }_{\mathrm{IR}}.$$

$${\text{SS}}_q\equiv \frac{q{\mathcal{F}}_1-{\mathcal{F}}_q}{1-q},{\mathcal{F}}_q=q{\mathcal{F}}_1+(q-1){\text{SS}}_q.$$

$${\mathcal{F}}_q\to {\mathcal{F}}_q+k_{1}q+k_2\to {\text{SS}}_q\to {\text{SS}}_q+k_2.$$

$${\text{SS}}_1={\mathcal{F}}_1^{\prime }-{\mathcal{F}}_1$$

$$d=4:{\text{SS}}_1=-4\mathrm{a},d=6:{\text{SS}}_1=-96\mathrm{a},$$

$$C_{T,d}=\frac{(-1{)}^{\frac{d}{2}}(d+1)!}{(d-1)[(\frac{d}{2}-1)!{]}^2}{\mathcal{F}}_1^{\prime \prime },{\text{SS}}_1^{\prime }=\frac{1}{2}{\mathcal{F}}_1^{\prime \prime }.$$

Full text

Imperial-TP-AT-2017-05,      NSF-KITP-17-061

1 Introduction

The conformal anomaly of a classically Weyl invariant theory in 6d depends on 4 independent coefficients ${\rm a},{\rm c}_{1},{\rm c}_{2},{\rm c}_{3}$ Bonora:1985cq ; Deser:1993yx ; Bastianelli:2000hi ; Boulanger:2007ab

[TABLE]

where $E_{6}$ is the 6d Euler density and the three Weyl invariants are $I_{1}=C_{\alpha\mu\nu\beta}C^{\mu\rho\sigma\nu}C_{\rho\alpha\beta\sigma}$,    $I_{2}=C_{\alpha\beta\mu\nu}C^{\mu\nu\rho\sigma}C_{\rho\sigma\alpha\beta},$  $I_{3}=C_{\mu\nu\alpha\beta}\nabla^{2}C^{\mu\nu\alpha\beta}+...$. As (1) appears in the log UV divergent part of the effective action, ${\rm c}_{3}$ can be determined from the 2-point function of stress tensor $\langle TT\rangle$, ${\rm c}_{2}$ and ${\rm c}_{3}$ – from the 3-point function222 $\langle TTT\rangle$ in 6d depends on three parameters, but one of them is related to the 2-point function or ${\rm c}_{3}$ by a conformal Ward identity Osborn:1993cr ; Erdmenger:1996yc ; Bastianelli:1999ab . and the ${\rm a}$-coefficient – from the 4-point function.

In the presence of $(1,0)$ supersymmetry one expects that the Weyl invariants $I_{i}$ are bosonic parts of only two possible 6d superinvariants, i.e. the coefficients ${\rm c}_{i}$ should satisfy one linear relation. As discussed in Beccaria:2015ypa , free-theory calculations Bastianelli:2000hi , strong-coupling (holography) arguments Kulaxizi:2009pz , and studies in other contexts Hofman:2008ar ; Safdi:2012sn ; Bueno:2015lza indicate that this relation is333In the case of $(2,0)$ supersymmetry, the three invariants $I_{i}$ are parts of a single superinvariant (6d conformal supergravity action Beccaria:2015uta ; Butter:2016qkx ; Butter:2017jqu ) and thus ${\rm c}_{i}$ obey the additional constraint ${\rm c}_{1}-4{\rm c}_{2}=0$. Then there is only one independent ${\rm c}$-coefficient: ${\rm c}_{1}=4{\rm c}_{2}=-12{\rm c}_{3}$. This relation holds for the free $(2,0)$ tensor multiplet Bastianelli:2000hi as well as for the large $N$ strong coupling limit of the interacting $(2,0)$ theory described by supergravity in AdS7 Henningson:1998gx .

[TABLE]

The 6d chiral ($SU(2)$ R-symmetry and gravitational) anomalies are encoded in the 8-form polynomial parametrized by 4 numerical coefficients $(\alpha,\beta,\gamma,\delta)$.444Explicitly, \mathcal{I}_{8}=\textstyle\frac{1}{4!}\big{(}\alpha\,c_{2}^{2}+\beta\,c_{2}\,{\rm p}_{1}+\gamma\,{\rm p}_{1}^{2}+\delta\,{\rm p}_{2}\big{)}\ ,\ \ \ \ c_{2}={\rm tr}\,F^{2}\ ,\ \ \ {\rm p}_{1}=-\tfrac{1}{2}\,{\rm tr}\,R^{2},\ \ \ \textstyle{\rm p}_{2}=-\frac{1}{4}\,{\rm tr}\,R^{4}+\frac{1}{8}\,({\rm tr}\,R^{2})^{2}\ . The 6d chiral and Weyl anomalies belong to a supersymmetry multiplet Howe:1983fr ; Manvelyan:2000ef ; Manvelyan:2003gc and as in the 4d case Buchbinder:1986im ; Anselmi:1997ys ; Aharony:2007dj ; Kuzenko:2013gva one expects to find linear relations between their coefficients, i.e. between $({\rm a},{\rm c}_{1},{\rm c}_{2},{\rm c}_{3})$ and $(\alpha,\beta,\gamma,\delta)$. Ref. Cordova:2015fha derived such relation for the ${\rm a}$-coefficient using supersymmetry and the results from the background supergravity couplings 555In our normalization the a-anomaly of (2,0) tensor multiplet is ${\rm a}(T^{(2,0)})=-{7\over 1152}$.

[TABLE]

The 4 coefficients in (3) could have been be fixed also from the ${\rm a}$-anomalies for the 4 multiplets: free tensor, free hyper, the (2,0) multiplet at large $N$ and the higher derivative vector multiplet (the ${\rm a}$-anomaly of which was found in Beccaria:2015uta ; Beccaria:2015ypa after Cordova:2015fha already appeared).

Assuming that similar linear relations exist also for ${\rm c}_{1},{\rm c}_{2},{\rm c}_{3}$, in Beccaria:2015ypa we attempted to fix their form using the available data about ${\rm c}$-anomalies of particular (1,0) superconformal theories. The linear relations for ${\rm c}_{1}$ and ${\rm c}_{2}$ in terms of $(\alpha,\beta,\gamma,\delta)$ contain, in general, 8 coefficients (${\rm c}_{3}$ is given by (2)). We first used the values of anomaly coefficients for free scalar $S^{(1,0)}$ and tensor $T^{(1,0)}$ multiplets

[TABLE]

built out of the standard 2-derivative real scalar $\varphi$, Majorana-Weyl (MW) spinor $\psi$ and (anti) selfdual rank 2 tensor $T$ with known Weyl anomalies Bastianelli:2000hi . This gave 4 coefficients out of 8. One more coefficient was fixed by considering the 4-derivative vector multiplet $V^{(1,0)}$ (see (8) below) on a Ricci-flat background when its Weyl anomalies can be readily computed. Two more coefficients were found from the known anomalies of interacting (2,0) $A_{N}$ theory (see Beem:2014kka ; Cordova:2015vwa and refs. there). As a result, we were able to find the form of the relations for ${\rm c}_{i}$ in terms of $(\alpha,\beta,\gamma,\delta)$ up to one undetermined parameter $\xi$, i.e.

[TABLE]

with ${\rm c}_{3}$ given by (2). We then conjectured that the value of $\xi$ should be

[TABLE]

This particular choice was motivated Beccaria:2015ypa by certain special features in the rank dependence of the c-anomaly of particular interacting $(1,0)$ superconformal theories and also by potential relation between 6d and 4d anomalies for certain (1, 0) theories compactified on 2-torus Ohmori:2015pua .

Recently, the same expressions for ${\rm c}_{i}$ (1),(2) but with a different value of $\xi$

[TABLE]

were found in Yankielowicz:2017xkf in a different approach using the assumption that the supersymmetric Rényi entropy for (1,0) superconformal 6d theory should be is a cubic polynomial in inverse of its argument.

In this paper we will settle the question about the right value of $\xi$ in our original approach of Beccaria:2015ypa by using an additional information about the anomalies of the free $(1,0)$ vector supermultiplet. We will confirm that the value (7) suggested in Yankielowicz:2017xkf is indeed the correct one.

This multiplet is the higher-derivative (non-unitary) superconformal 6d $(1,0)$ vector multiplet $V^{(1,0)}$ that contains the 4-derivative gauge vector $V^{(4)}_{\mu}$ (with action $\sim\int F_{\mu\nu}\partial^{2}F^{\mu\nu}$), the 3-derivative MW spinors $\psi^{(3)}$ and the 2-derivative scalars $\varphi$ Ivanov:2005qf ; Beccaria:2015uta ; Beccaria:2015ypa (cf. (4))

[TABLE]

The anomaly polynomial for this multiplet has coefficients666The corresponding a-anomaly is $\ {\rm a}(V^{(1,0)})=-\frac{251}{210}$ Beccaria:2015uta ; Beccaria:2015ypa , in agreement with (3).

[TABLE]

so that using (1),(2), we should thus expect to find

[TABLE]

The direct computation of Weyl anomalies ${\rm c}_{i}$ for $V^{(1,0)}$ on a general curved background is challenging as it requires the knowledge of 6d Seeley-DeWitt coefficients for the corresponding higher derivative vector and spinor operators. In the two special cases – of a sphere and a Ricci flat space – that were discussed in Beccaria:2015uta ; Beccaria:2015ypa the higher derivative operators factorize and the anomalies can be readily computed using the expressions for the Seeley-DeWitt coefficients of 2nd order Laplacians. This fixes 3 out of 4 coefficients in (1) and thus does not allow to determine $\xi$. In fact, the higher derivative scalar, vector and (squared) spinor operators discussed below factorize also on a general Einstein space $R_{\mu\nu}={1\over d}g_{\mu\nu}R$ (on which the curvature invariants in (1) remain independent) and thus their 6d anomalies may be computed using the 6d Seeley coefficient of 2nd order Laplacians as was done in Bastianelli:2000hi for the standard scalar, spinor and 2-form fields.777We thank D. Diaz for this remark. We will use this observation below in Section 6.

To fix the value of $\xi$ it is sufficient, according to (10), to compute just ${\rm c}_{3}$ which itself is determined by the coefficient $C_{T}$ appearing in the 2-point function of stress tensor in flat background. In fact, as $C_{T}$ for the scalar and the 4-derivative vector is already known Osborn:2016bev ; Giombi:2016fct , it remains only to compute it for the 3-derivative spinor field.888We thank Ying-Hsuan Lin and Chi-Ming Chang for this suggestion.

In more detail, the 6d Weyl anomaly coefficient ${\rm c}_{3}$ in (1) is given by999In 4 dimensions the coefficient ${\rm c}$ of the Weyl-squared term in the trace anomaly is given by ${\rm c}={1\over 160}C_{T,4}$.

[TABLE]

where $C_{T,d}$ is the coefficient in the 2-point function of stress tensor in a $d$-dimensional CFT

[TABLE]

The coefficient $C_{T,d}$ is known for several unitary and non-unitary conformal theories Osborn:1993cr ; Petkou:1994ad ; Petkou:1995vu ; Buchel:2009sk ; Diab:2016spb ; Giombi:2016fct ; Osborn:2016bev . In particular, for the standard real conformal scalar and spin $1/2$ fermion one has

[TABLE]

where $n_{f}$ is the (complex) dimension of the spinor space ($n_{f}=2^{{d\over 2}-1}$ for Majorana and $2^{{d\over 2}-2}$ for MW case). For example, for a 4d Majorana fermion $C_{T,4}(\psi)=4$, while for a 6d MW fermion $C_{T,6}(\psi)=6$.101010Let us add also that for an (anti) self-dual 6d tensor one finds $C_{T,6}(T^{+})=54$. The corresponding values of ${\rm c}_{3}$ in (11) are ${\rm c}_{3}(\varphi)=\tfrac{1}{2520},\ \ {\rm c}_{3}(\psi)=\tfrac{1}{504},\ \ {\rm c}_{3}(T^{+})=\tfrac{1}{56}.$

Using the scalar value in (13) and the known value of $C_{T}$ for the 4-derivative gauge vector $V^{(4)}$ Osborn:2016bev ; Giombi:2016fct

[TABLE]

we find that $C_{T}$ for the 4-derivative vector multiplet (8) is given by

[TABLE]

Comparing this to (10), (11) we conclude that the two suggested values of $\xi$ in (6) and (7) correspond to

[TABLE]

As we shall find below, it is the second value (17) that is the correct result for the $C_{T}$ of the 3-derivative 6d MW fermion.

To find $C_{T}$ for a free conformal field one may follow the standard route of first determining the explicit form of the stress tensor $T_{\mu\nu}$ as a conformal primary or obtaining it from the metric variation of a Weyl-invariant action in curved background and then using (12).111111Some simplification is that in computing $\langle TT\rangle$ in (12) one can drop total derivative (descendant) parts in one of the two $T$-factors Osborn:2016bev . An alternative approach that we shall follow below is to exploit the relation between $C_{T}$ and the Rényi entropy Perlmutter:2013gua . As we shall demonstrate, this second approach turns out to be more efficient in the case of the higher-derivative conformal fields.

Given a CFT in flat even-dimensional space one has the following relation between the first derivative of the Rényi entropy $\SS_{q}$ (which is a function of $q$ defined in the next section) at $q=1$ and the coefficient $C_{T,d}$ in (12) Perlmutter:2013gua

[TABLE]

Here ${\rm V}_{S^{d}}$ is the volume of the sphere as in (12) and $\overline{{\rm V}}_{\mathbb{H}^{d-1}}$ is the finite coefficient in the regularized volume of the odd-dimensional unit-radius hyperbolic space $\mathbb{H}^{d-1}$ (${\Lambda_{\rm IR}}$ is an IR cutoff)

[TABLE]

In particular,

[TABLE]

and thus

[TABLE]

We shall start in section 2 with defining the Rényi entropy in terms of the free energy ${\cal F}_{q}$ on $S^{1}_{q}\times{\mathbb{H}}^{d-1}$, i.e. the product of a thermal circle (with length $\beta=2\pi q$) and hyperbolic space, thus relating $C_{T}$ to the second derivative of the free energy at $q=1$. We will then describe our method of computing this free energy using heat kernel representation.

To illustrate this method of computing free energy and $C_{T}$ in section 3 we will consider the examples of the 4- and 6-derivative conformal scalars in even number of dimensions. In section 4 we will discuss the case of the 4-derivative conformal gauge vector in 6d reproducing the value (14) of its $C_{T}$ obtained earlier by other methods.

Section 5 will contain a similar computation of free energy and thus Rényi entropy and $C_{T}$ for the 3-derivative conformal fermion in $d=4$ and $d=6$. For higher derivative operators the computation of $C_{T}$ turns out to be subtle: surprisingly, a naive approach (discussed in Appendix A) leads to the value in (16) while the correct evaluation gives (17).

In section 6 we will provide an independent derivation of the conformal anomalies of the vector multiplet (10) with $\xi$ given by (7) by directly computing the Seeley-DeWitt coefficients of the higher derivative operators involved using the fact of their factorization on a generic Einstein background.

In Appendix A we will supplement the discussion in section 5 by explaining a different method of computing the free energy on $S^{1}_{q}\times{\mathbb{H}}^{d-1}$. In Appendix B we will compute $C_{T}$ for the non-unitary 2-derivative conformal vector theory which has no gauge invariance in $d\not=4$. Finally, in Appendix C, we shall present the result for the conformal anomalies for a family of vector multiplets generalizing (8) that shows again the agreement with the relations (2), (1) with $\xi$ given by (7).

2 Free energy for conformal fields on $S^{1}_{q}\times\mathbb{H}^{d-1}$

and Rényi entropy

The Rényi entropy $\SS_{q}$ is a measure of generalized quantum entanglement and can be computed from traces of the reduced density matrix raised to a power $q\geq 0$. For a $d$-dimensional CFT, the Rényi entropy across $S^{d-2}$ may be equivalently extracted from the partition function on $q$-cover of the sphere $S^{d}$ or from the thermal partition function on $S^{1}_{q}\times\mathbb{H}^{d-1}$ (see Casini:2011kv ; Klebanov:2011uf ; Lewkowycz:2014jia and refs. there).121212The metrics of the two spaces are related by a singular conformal rescaling

ds^{2}_{q\rm-sphere}=\sin^{2}\theta\,q^{2}d\tau^{2}+d\theta^{2}+\cos^{2}\theta\ d\Omega^{2}_{d-2}=\sin^{2}\theta\,\big{(}q^{2}d\tau^{2}+d\rho^{2}+\sinh^{2}\rho\,d\Omega^{2}_{d-2}\big{)}=\cosh^{-2}\rho\,ds^{2}_{S^{1}_{q}\times{\mathbb{H}}^{d-1}}

Here $\tau\in(0,2\pi)$ and $\sinh\rho=\cot\theta$. This transformation maps the subspace $S^{d-2}$ to the boundary of $\mathbb{H}^{d-1}$. For $q=1$ the space $S^{1}_{q}\times\mathbb{H}^{d-1}$ becomes conformal to regular $S^{d}$ and thus also to $\mathbb{R}^{d}$ as ds^{2}=dz^{2}+z^{2}dx_{0}^{2}+dx_{n}dx_{n}=z^{2}\big{(}dx_{0}^{2}+{{dz^{2}+dx_{n}x_{n}}\over z^{2}}\big{)}.

Here $\mathbb{H}^{d-1}$ is real hyperbolic space (of curvature radius $r=1$) and the length of the thermal circle $x_{0}=q\tau$ or the inverse temperature is $\beta=2\pi q$.

2.1 General relations

Here we shall use the latter definition of $\SS_{q}$ in terms of the partition function or free energy on $S^{1}_{q}\times\mathbb{H}^{d-1}$ for even $d$. Given a free real conformal field $\Phi$ with the action

[TABLE]

where ${\cal O}$ is a (possibly higher order) covariant differential operator including curvature terms needed to ensure the Weyl invariance of (22) in a general curved background, the corresponding free energy on ${S^{1}_{q}\times\mathbb{H}^{d-1}}$ is

[TABLE]

In the present case of a homogeneous space $F_{q}$ is proportional to its volume, i.e. to $2\pi q\,{\rm V}_{{\mathbb{H}}^{d-1}}$ in (19). Extracting the IR divergent factor, we may define the IR finite "free energy" ${\mathcal{F}}_{q}$ by

[TABLE]

For even $d$ the free energy on ${S^{1}_{q}\times\mathbb{H}^{d-1}}$ does not contain logarithmic UV divergences131313 Since $S^{1}_{q}$ factor is flat and $\mathbb{H}^{d-1}$ is conformally flat, all logarithmic divergent terms containing the Weyl tensor vanish, while the Euler density in $d$ dimensions vanishes when evaluated on $\mathbb{H}^{d-1}$. while the non-universal power divergent part of ${\mathcal{F}}_{q}$ (which is proportional to the volume and is thus linear in $q$) should be subtracted using some regularization prescription.

The finite Rényi entropy is then given by

[TABLE]

Note that under a linear in $q$ and constant shift of the free energy we have

[TABLE]

As all power UV divergent terms in ${\mathcal{F}}_{q}$ are linear in $q$ they drop out of $\SS_{q}$ which is thus UV finite. The $q=1$ value of the Rényi entropy which is the entanglement entropy

[TABLE]

is sensitive to the constant ($q$-independent) part of ${\mathcal{F}}_{q}$. $\SS_{1}$ is expected to be proportional to the ${\rm a}$-anomaly coefficient of the $d$-dimensional CFT, e.g.,141414In 4 dimensions (cf. (1)) $(4\pi)^{2}T^{\mu}_{\mu}=-{\rm a}\,R^{*}R^{*}+{\rm c}\,C^{\mu\nu\lambda\rho}C_{\mu\nu\lambda\rho}$.

[TABLE]

as that happens when $F_{d}$ is computed on the $q$-cover of the sphere $S^{d}$ Solodukhin:2008dh ; Dowker:2010bu ; Dowker:2010nq ; Solodukhin:2010pk ; Casini:2011kv ; Aros:2014xga .151515One expects that the log UV divergent part of free energy on $q$-cover of the $S^{d}$ should be matching the log IR part of free energy on $S^{1}_{q}\times\mathbb{H}^{d-1}$, and that was checked on specific examples, though a general proof of this statement appears to be missing in the literature. However, the transformation between the $q$-cover of the $S^{d}$ and $S^{1}_{q}\times\mathbb{H}^{d-1}$ is a non-trivial Weyl rescaling (cf. footnote 12) and thus the two free energies may a priori differ by a Weyl-anomaly term. It was observed that for fields with gauge invariance $\SS_{1}$ computed on ${S^{1}_{q}\times\mathbb{H}^{d-1}}$ is not automatically proportional to the Weyl anomaly a-coefficient (see Donnelly:2014fua ; Huang:2014pfa for 4d vectors and Nian:2015xky for 6d antisymmetric tensors), but one can achieve this by shifting ${\mathcal{F}}_{q}$ by a constant (that may be interpreted as an edge mode contribution).

The $C_{T}$ coefficient which is proportional to the first derivative of the Rényi entropy (18) may be expressed in terms of the second derivative of the free energy ${\mathcal{F}}_{q}$ and thus is not sensitive to the shifts in (26). Explicitly,

[TABLE]

In particular (see (21),(11) and 9)

[TABLE]

Thus to compute $C_{T}$ we need to find the free energy ${\mathcal{F}}_{q}$ on $S^{1}_{q}\times\mathbb{H}^{d-1}$.

2.2 Computational scheme

The covariant kinetic operator ${\cal O}$ specified to ${S}^{1}_{q}\times X^{d-1}$ where $X^{d-1}$ is a symmetric space like $S^{d-1}$ or ${\mathbb{H}}^{d-1}$ will be a polynomial in derivatives $\partial_{0}$ along the "euclidean time" direction ${S}^{1}$ and the covariant derivatives $D_{i}\equiv\bm{D}_{i}$ on $X^{d-1}$, i.e. symbolically ${\cal O}=P(i\,\partial_{0},-\bm{D}^{2})$ (with $X^{d-1}$ curvature factors translating into the coefficients of lower-order terms in $P$). In the case of $X^{d-1}={\mathbb{H}}^{d-1}$ the free energy ${\mathcal{F}}_{q}$ in (23),(24) will have the following structure

[TABLE]

where $\frac{n}{q}$ is the eigenvalue of $i\partial_{0}$ and $d\mu(\lambda)$ is the spectral measure for the continuous eigenvalue $\lambda$ of the spatial operator $-\bm{D}^{2}+...$ (a particular definition of $\lambda$ will depend on a type of the field $\Phi$ in (22), see below). The summation index $n$ takes values in $\mathbb{Z}$ for bosons and in $\mathbb{Z}+\frac{1}{2}$ for fermions.161616The antiperiodicity of fermions in ”thermal” circle is related to the original definition of partition function on $q$-cover of $S^{d}$.

It turns out that for conformal fields the kinetic operators ${\cal O}$ restricted to ${S}^{1}_{q}\times X^{d-1}$, i.e. $P(i\,\partial_{0},-\bm{D}^{2})$, have special factorized structure, i.e. are given by a product of simple two-derivative factors.171717This applies to bosonic operators and squared fermionic operators. A particular reason for this can be understood by observing that the operators on ${S}^{1}\times\mathbb{H}^{d-1}$ and ${S}^{1}\times S^{d-1}$ are formally related by an analytic continuation changing the sign of the curvature. The thermal partition function on ${S}^{1}\times S^{d-1}$ is expressed in terms of characters of conformal group and this in turn is related to factorization of the (higher-derivative) kinetic operator discussed in detail in Beccaria:2014jxa . In the case of ${S}^{1}_{q}\times S^{d-1}$ we get

[TABLE]

where the sum over $m$ is over the discrete spectrum of $-\bm{D}^{2}+...$ on $S^{d-1}$ and $\mu(m)$ is the multiplicity factor of the eigenvalue with label $m$. The higher-derivative Weyl-covariant operators ${\cal O}=D^{2p}+...$ turn out to factorize Beccaria:2014jxa into simple factors so that the corresponding eigenvalues on $S^{1}_{q}\times S^{d-1}$ are

[TABLE]

where $r$ is the radius of $S^{d-1}$. In this case, the standard free energy $F_{q}$ in (23) is expressed in terms of the single-particle partition function $\mathcal{Z}(x)$ that has a simple structure

[TABLE]

Here $m+\ell_{k}$ correspond to the single-particle energies or integer dimensions of conformal operators in $\mathbb{R}^{d}$ built out of $\Phi$ and its derivatives.

The factorization of the higher-derivative Weyl-covariant kinetic operator ${\cal O}$ on ${S}^{1}\times\mathbb{H}^{d-1}$ is thus intimately related to its factorization on ${S}^{1}\times S^{d-1}$ which in turn is related to integrality of dimensions of the CFT operators in $\mathbb{R}^{d}$.181818Similar factorization is found also for ${\cal O}$ defined on $S^{d}$ or ${\mathbb{H}}^{d}$.

One may also consider the analytic continuation between $S^{d-1}$ and ${\mathbb{H}}^{d-1}$ at the level of the spectrum (see Camporesi:1994ga ; Gopakumar:2011qs and Appendix C of Beccaria:2014xda ). For example, for a 2nd order Laplacian acting on symmetric traceless rank $s$ tensors on a homogeneous space one has the following spectrum on $S^{d-1}$ with radius $r$

[TABLE]

The eigenvalues $\omega_{\lambda}$ of the same operator on $\mathbb{H}^{d-1}$ with curvature radius $r$ are obtained by replacing

[TABLE]

Here $0\leq\lambda<\infty$ is the eigenvalue of the following operator on $\mathbb{H}^{d-1}$ (here and in what follows we set the radius of ${\mathbb{H}}^{d-1}$ to be $r=1$) Camporesi:1994ga

[TABLE]

The analytical continuation (36) then translates the factorization (33) into the one on $S^{1}\times\mathbb{H}^{d-1}$.

In addition, we need to replace the sum $\sum_{m}\mu(m)$ in (32) by $\int d\mu(\lambda)$ in (31) with a definite correspondence between the discrete multiplicity on $S^{d-1}$ and the spectral measure on ${\mathbb{H}}^{d-1}$. The latter is the Plancherel measure for the transverse traceless symmetric rank $s$ field on ${\mathbb{H}}^{d-1}$ corresponding to the spectrum (37) Camporesi:1994ga

[TABLE]

Having ${\cal O}$ factorized into a product of second-derivative factors, the polynomial $P_{\mathbb{H}}$ in (31) may be written in the product form which is the counterpart of (33),

[TABLE]

where $\alpha_{k}$ are real constants (appearing in $\pm$ conjugate pairs so that $P_{\mathbb{H}}$ is real). Then $\log P_{\mathbb{H}}$ in (31) becomes the sum of $p$ terms. Using the proper-time representation separately for each log term in the sum we then get (in bosonic case)

[TABLE]

Here $K_{S^{1}}$ is the trace of the heat kernel of $-\partial_{0}^{2}$ on $S^{1}$ while $\overline{K}_{{\mathbb{H}}^{d-1}}(t;\alpha)$ may be interpreted as the heat kernel corresponding to the operator \big{(}\sqrt{{\bm{\Delta}}_{s}}+i\alpha\big{)}^{2} on ${\mathbb{H}}^{d-1}$ (cf. (37)). Using the Poisson resummation191919 In general,

$\sum_{n\in\mathbb{Z}}e^{-t{(n+\gamma)^{2}\over q^{2}}}=\frac{2\,\pi\,q}{(4\,\pi\,t)^{1/2}}\sum_{n\in\mathbb{Z}}\,e^{-\frac{\pi^{2}\,n^{2}\,q^{2}}{t}+2\pi i\gamma n}\ \ .$

we may represent $K_{S^{1}}(t)$ as

[TABLE]

Similarly, in the fermion (antiperiodic) case one finds

[TABLE]

Assuming $t>0$ the integral over $\lambda$ in (41) is convergent, i.e. the relevant real part of $\overline{K}_{{\mathbb{H}}^{d-1}}(t;\alpha)$ is proportional to a finite polynomial in $t$, i.e.

[TABLE]

where $\nu_{j}$ are numerical constants depending on $\alpha$, dimension $d$ and spin of the field. The integral over $t$ in (40) is then power-divergent at $t=0$ for $n=0$ term in (42) or (43). Subtracting these power divergences as a proper-time regularization prescription corresponds to omitting the $n=0$ term in the sum. As a result, we are left with a finite sum over $n\geq 1$ expressing ${\mathcal{F}}_{q}$ as a finite polynomial in $q^{-1}$ with coefficients proportional to the Riemann zeta-function values.202020 In the antiperiodic case one has

$\zeta_{2k}=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}\quad\to\quad\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2k}}=(2^{1-2k}-1)\,\zeta_{2k}\ .$

To summarize, the computation of the free energy ${\mathcal{F}}_{q}$ will contain the following sequence of steps: (i) integration over the eigenvalue $\lambda$; (ii) integration over the proper time $t$ with $t\to 0$ power divergences subtracted; (iii) performing the remaining finite sum over $n\not=0$. We shall illustrate this procedure in detail on several examples below. Having found ${\mathcal{F}}_{q}$ one can then compute the Rényi entropy in (25) and $C_{T}$ in (29).

3 Scalar fields

To illustrate the relation (18),(29) in this section we will use it compute $C_{T}$ for free higher-derivative conformal scalar theories in even dimension $d$, reproducing the results obtained previously by other methods in a novel way.

3.1 $\partial^{2}$ scalar

The standard action for the conformally coupled scalar is

[TABLE]

The corresponding free energy on $S^{1}_{q}\times\mathbb{H}^{d-1}$ is given by ($R({\mathbb{H}}^{d-1})=-(d-1)(d-2)$)

[TABLE]

The spectrum of the operator ${\bm{\Delta}_{0}}$ (i.e. the $s=0$ case of (37)) is $\frac{n^{2}}{q^{2}}+\lambda$ where $n\in\mathbb{Z}$ and $\lambda\geq 0$. The spectral measure is given by the $s=0$ case of (38), in particular, in $d=4$ and $d=6$,

[TABLE]

In $d=4$ we get from (41)

[TABLE]

Then using (42),(20) we find

[TABLE]

where we omitted the $n=0$ mode which corresponds to subtracting the $\Lambda^{4}$ UV divergence ($t=\varepsilon=\Lambda^{-2}\to 0$). The resulting Rényi entropy and the Weyl anomaly coefficients have indeed the standard values (see (30))

[TABLE]

Similarly, in $d=6$

[TABLE]

where we again dropped the $n=0$ term in the sum corresponding to subtracting the $\Lambda^{6}$ and $\Lambda^{4}$ UV divergences. The above values for $C_{T,d}$ are in agreement with the general expression in (13).

3.2 $\partial^{4}$ scalar

The Weyl-invariant action for the 4-derivative scalar in curved 4d space is given by Fradkin:1981jc

[TABLE]

The generalization of the $D^{4}$ operator in (54) to any $d>4$ is the Paneitz operator Paneitz:1983

[TABLE]

We did not write explicitly the $D^{2}R$ term as we will be interested in the homogeneous $S^{1}\times X^{d-1}$ background with

[TABLE]

where we introduced the curvature sign factor $\epsilon$ which is $+1$ for $X^{d-1}={\mathbb{H}}^{d-1}$ and $-1$ for $X^{d-1}=S^{d-1}$. Then (55) is found to factorize in either of the following two $d$-independent ways

[TABLE]

where $\bm{D}^{2}\equiv D^{i}D_{i}$ and ${\bm{\Delta}_{0}}=-\bm{D}^{2}-\tfrac{(d-2)^{2}}{4}\epsilon$ is the conformal scalar Laplacian as in (46). This factorization was already observed on ${S}^{1}\times S^{d-1}$ where $\epsilon=-1$ (see eq. (B.22) in Beccaria:2014xda for $d=4$).

The eigenvalues of ${\cal O}^{(4)}$ are thus naturally expressed in terms of the eigenvalue $\lambda$ of the conformal scalar Laplacian on ${\mathbb{H}}^{d-1}$ in (37)

[TABLE]

This is thus the special case of (39) with $\alpha_{k}=\pm 1$ so that the corresponding free energy can be computed as in (40)–(2.2). Explicitly, we find that in this case ${\overline{K}}_{{\mathbb{H}}^{d-1}}(t)$ is given by (2.2) with $\alpha=1$ so that for $d=4$ (cf. (49),(50))

[TABLE]

These values of the Weyl anomaly coefficients ${\rm a}$ and ${\rm c}$ for the 4-derivative scalar agree with the result of the direct computation in Fradkin:1981jc ; Fradkin:1985am .

In $d=6$ get (cf. (51)–(53))

[TABLE]

The value of ${\rm a}$ in (65) agrees with the one found in Beccaria:2015uta (see Table 1 there).

The above values of $C_{T}$ in (62) and (65) are in agreement with the general expression for the 4-derivative conformal scalar in dimension $d$ found in Guerrieri:2016whh ; Osborn:2016bev

[TABLE]

3.3 $\partial^{6}$ scalar

The general expression for the Weyl-covariant 6-derivative scalar operator in curved background can be found, e.g., in Osborn:2015rna . Ignoring terms with derivatives of the curvature and specifying to $d=6$ it can be written as

[TABLE]

where the Schouten tensor $P_{\mu\nu}$ and its trace $P$ are in general defined as

[TABLE]

Using the properties (56) of the curvature of ${S}^{1}\times{\mathbb{H}}^{5}$ we find

[TABLE]

Like the 4-derivative scalar operator (57),(58) (where $\epsilon=1$ for ${\mathbb{H}}^{d-1}$) ) this operator may be factorized in the two possible ways

[TABLE]

so that the corresponding eigenvalues are given by (39) with $\alpha_{1}=0,\ \alpha_{2}=2,\ \alpha_{3}=-2$. We thus get a combination of the standard 2-derivative scalar and a conjugate pair of operators with the shift parameter $\alpha=2$. The heat kernel for the latter is given by (2.2) and as a result we find in $d=6$ (cf. (63)–(65))

[TABLE]

The value of ${\rm a}$-coefficient agrees with the one following from the partition function of 6-order GJMS operator on $S^{6}$ Beccaria:2015vaa while the value of $C_{T,6}$ agrees with the $d=6$ case of the general expression for $\partial^{6}$ conformal scalar in Osborn:2016bev

[TABLE]

4 Conformal vector fields

Conformal generalization of the Maxwell theory to general dimension $d$ has a higher derivative Lagrangian $L=F_{\mu\nu}(\partial^{2})^{d-4\over 2}\,F^{\mu\nu}$. In particular, in 6 dimensions this gives a 4-derivative non-unitary vector gauge theory that we shall consider below. The computation of $C_{T}$ for 2-derivative non gauge invariant conformal vector theory in generic $d$ (reducing to Maxwell theory for $d=4$) will be discussed in Appendix B.

4.1 $\partial^{2}$ gauge vector in $d=4$

It useful to start with recalling the computation of free energy of the Maxwell theory on $S^{1}_{q}\times{\mathbb{H}}^{3}$. The closely related case of $S^{1}_{q}\times S^{3}$ background was discussed, e.g., in section 2.2 of Beccaria:2014jxa . Starting with $I=-{1\over 4}\int d^{4}x\sqrt{g}\,F^{\mu\nu}F_{\mu\nu}$ where $F_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ and fixing the $V_{0}=0$ gauge one ends up with the free energy expressed in terms of the operator defined on transverse 3-vector (cf. (46))

[TABLE]

where ${\bm{\Delta}_{1}}$ is the $d=4,\,s=1$ case of the operator in (37) with the eigenvalue $\lambda$. The corresponding spectral density is the $s=1$ case of (38), i.e. in $d=4$ and $d=6$ it reads

[TABLE]

As a result, the ${\mathbb{H}}^{3}$ part of heat kernel in (40) is (cf. (48))

[TABLE]

and thus integrating over $t$, dropping quartic and quadratic divergences and summing over $n$ as in (49) we get

[TABLE]

This reproduces the correct value of $C_{T}$ or ${\rm c}$-coefficient for the Maxwell field but not the standard value of the a-coefficient that should be $\frac{31}{180}=\frac{4}{45}+{\frac{1}{12}}$. As mentioned in section 2, this matching need not be expected to follow automatically when free energy is computed on $S^{1}_{q}\times{\mathbb{H}}^{3}$ but one can formally enforce the relation between the $\SS_{1}$ and the Weyl anomaly ${\rm a}$-coefficient by shifting ${\mathcal{F}}_{q}$ and thus $\SS_{q}$ by a constant as in (26):

[TABLE]

4.2 $\partial^{4}$ gauge vector in $d=6$

Defined on a curved background, the 6d conformal vector gauge theory has the following Weyl-invariant action Beccaria:2015uta

[TABLE]

where $F_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$. To compute the corresponding free energy on ${S}^{1}_{q}\times{\mathbb{H}}^{d-1}$ it is convenient to choose again the temporal gauge $V_{0}=0$. This leads to the ghost factor $(\det\partial_{0}^{2})^{1/2}$ in the partition function. Using (56) the Lagrangian in (82) then becomes (here $i,j,...=1,...,5$ are indices of $\mathbb{H}^{5}$)

[TABLE]

Change of the variables $V_{i}\to(V_{i}^{\perp},\varphi)$,

[TABLE]

introduces the Jacobian factor $(\det\bm{D}^{2})^{1/2}$  ($\bm{D}^{2}\equiv D^{i}D_{i}$) in the path integral, while the Lagrangian (83) takes the form

[TABLE]

where $\bm{\Delta}_{1}$ is the $d=6$ case of the operator defined in (37). Integrating over $\varphi$ we get a factor \big{[}\det(\partial_{0}^{2}\bm{D}^{2})\big{]}^{-1/2} (which cancels against the the previously mentioned ghost and Jacobian factors) as well as the contribution of the conformal 6d scalar \big{[}\det(-\partial^{2}_{0}-\bm{D}^{2}-4)\big{]}^{-1/2} (see (46)).

The remaining 4-derivative operator acting on $V^{\perp}_{i}$ in (85) factorizes exactly as in the 4-derivative scalar case (57) (with $\bm{\Delta}_{0}\to\bm{\Delta}_{1}=\bm{\Delta}^{\prime}_{1}-1$)

[TABLE]

As in (57), the same factorization as in (87),(88) is found if one considers the theory (82) on $S^{1}\times S^{5}$.212121 Due to the change of the sign of the curvature of the spatial part, here $\bm{\Delta}^{\prime}_{1}=-\bm{D}^{2}+4$ that has discrete eigenvalues on the sphere, i.e. $\bm{\Delta}^{\prime}_{1}\to m^{2}+6m+8$ with integer $m\geq 0$. The $4\partial_{0}^{2}$ term in (87) here has flippped sign (as it came from the curvature term in (82)) and thus we find that on $S^{1}\times S^{5}$

\displaystyle\textstyle\big{(}-\partial_{0}^{2}+\bm{\Delta}^{\prime}_{1}\big{)}^{2}-4\partial_{0}^{2}\ \to\ \big{[}\big{(}\frac{n}{q}+i\big{)}^{2}+(m+3)^{2}\big{]}\,\big{[}\big{(}\frac{n}{q}-i\big{)}^{2}+(m+3)^{2}\big{]}=\big{[}\frac{n^{2}}{q^{2}}+(m+2)^{2}\big{]}\,\big{[}\frac{n^{2}}{q^{2}}+(m+4)^{2}\big{]}\ .

This leads to the thermal free energy corresponding to the spectrum of dimensions $w_{m}=m+2,\ m+4$ expected from the operator counting on $\mathbb{R}\times S^{5}$ (as explicitly discussed in Beccaria:2014jxa in the 4d case).

Using that $\bm{\Delta}_{1}$ in (37) has the eigenvalue $\lambda$, the polynomial $P_{\mathbb{H}}$ in (39) is again given by (59). The only difference compared to the 4-derivative 6d scalar case is that now the spectral measure is given by the $s=1$ case of (38), i.e. by $d\mu_{1,5}$ in (77).

As a result, the free energy ${\mathcal{F}}_{q}$ and thus the Rényi entropy and $C_{T}$ for the 4-derivative vector theory (82) is given by the sum of the contribution of the transverse spatial vector (with the kinetic operator ${\cal O}^{(4)}_{1}$) and of the standard 6d conformal scalar, i.e.

[TABLE]

The scalar contribution was already given in (51)–(53). The total ${\mathbb{H}}^{5}$ heat kernel factor in the resulting free energy is then (cf. (72))

[TABLE]

and thus finally

[TABLE]

The value of $C_{T,6}$ is the same (14) as quoted in the Introduction, found earlier by other methods in Osborn:2015rna ; Giombi:2016fct . To also reproduce the correct value ${\rm a}={275\over 8\times 7!}$ Beccaria:2015uta of the a-anomaly for the 4-derivative 6d vector field one needs, as in the $d=4$ vector case (81), to shift ${\mathcal{F}}_{q}$ and thus $\SS_{q}$ by the constant term  $-{14\over 45}$.

5 Fermionic fields

Finally, let us discuss the fermionic fields. We shall first review the computation of free energy and $C_{T}$ for the standard Dirac fermion and then consider the conformal 3-derivative fermion which is part of the 6d superconformal vector multiplet (8).

5.1 $\not{\partial}$ fermion

The curved space Weyl-invariant action for a standard massless fermion

[TABLE]

leads to the following formal expression for its free energy on ${S^{1}_{q}\times\mathbb{H}^{d-1}}$ in terms of the eigenvalues of the squared operator $(i\not{D})^{2}=-\partial_{0}^{2}+(i\not{\bm{D}})^{2}$ Camporesi:1992tm ; Camporesi:1995fb

[TABLE]

Here $n_{f}$ is the complex dimension of the spinor space (e.g., $n_{f}=2$ for a Weyl fermion in $d=4$ or MW fermion in $d=6$) and the sum over half-integer $n$ corresponds to the antiperiodic boundary conditions on the "thermal" circle $S^{1}$. $\lambda$ is the eigenvalue of the operator $(i\not{\bm{D}})^{2}=-\bm{D}^{2}+{1\over 4}R$ equal to $-\bm{D}^{2}-{1\over 4}(d-1)(d-2)$ on ${\mathbb{H}}^{d-1}$ (cf. (37)). The corresponding spin 1/2 Plancherel measure for even $d$ is Camporesi:1992tm ; Camporesi:1995fb (cf. (38))

[TABLE]

Using the proper time representation for the log in (94) we get as in (40),(43)

[TABLE]

The power UV divergences in the proper time integral should be again subtracted by omitting the $n=0$ term in the sum. Explicitly, one finds in $d=4$

[TABLE]

and in $d=6$

[TABLE]

Eqs. (101),(104) give the correct known values of the ${\rm a}$ and ${\rm c}$ Weyl anomaly coefficients in $d=4$ Duff:1977ay and in $d=6$ Bastianelli:2000hi and the values of $C_{T}$ also agree with the general expression for $C_{T,d}(\psi)$ given in (13). The expressions for the Rényi entropy agree with Klebanov:2011uf ; Nian:2015xky .

5.2 Conformal $\not{\partial}^{3}$ fermion

The Weyl-invariant operator for a 3-derivative fermion was first found in the context of extended conformal supergravity Bergshoeff:1980is in $d=4$ Fradkin:1981jc (for Majorana fermions)

[TABLE]

In $d=6$ the analogous 3-derivative operator was recently found in Butter:2017jqu 222222We thank D. Butter for pointing this out to us and a clarifying discussion.

[TABLE]

The generalization of (105),(106) to any $d$ reads

[TABLE]

where $P_{\mu\nu}$ is the Schouten tensor as in (68). On $S^{1}\times\mathbb{H}^{d-1}$ we can use (56) to get the following explicit form of (107)

[TABLE]

As a result, its square factorizes in a $d$-independent manner just like in the 4-derivative scalar case in (57) (cf. also (87))232323The factorization of the operator (105) on $S^{1}\times S^{3}$ was observed in Beccaria:2014xda .

[TABLE]

The corresponding eigenvalue polynomial (39) is then the product of the standard fermion part and the same factor as in the $\partial^{4}$ scalar case in (59) (and also has a similar structure as the result in the 4-derivative vector case in (88)).

Using the expression for the spin 1/2 spectral measure in (96) and starting with (97) we then find in $d=4$ (cf. (60)–(62))

[TABLE]

The values of ${\rm a}=-{3\over 80}$ and ${\rm c}=-{1\over 120}$ for a Majorana fermion ($n_{f}=2$) agree with the ones found by direct computation in Fradkin:1981jc ; Fradkin:1985am .242424In the notation of Table 6.1 in Fradkin:1985am one has for the 3-derivative $\Lambda$-spinor: $\beta_{1}={7\over 240},\ \beta_{2}=-{1\over 60}$ with ${\rm a}=-\beta_{1}+\tfrac{1}{2}\beta_{2},\ {\rm c}=\tfrac{1}{2}\beta_{2}$. To compare, for a real 4-derivative scalar $\beta_{1}={1\over 90}\ ,\ \beta_{2}=-{2\over 15}$, giving ${\rm a}=-{7\over 90},\ {\rm c}=-{1\over 15}$ in agreement with values given earlier in (62) (note that the $\partial^{4}$ scalar $\varphi$ in Table 6.1 is complex).

In 6 dimensions we get (cf. (63)–(65)) and (102)–(104))

[TABLE]

Thus for a 6d MW fermion with $n_{f}=2$ we get ${\rm a}=\tfrac{39}{32\times 7!}$ in agreement with the value found in Beccaria:2015uta while

[TABLE]

This confirms the value corresponding to $\xi_{{}_{{\rm YZ}}}$ in (17).

To emphasize that (117) is a result of a rather non-trivial computation, in Appendix A we shall present an alternative way of arriving at (117) based on the approach that does not use the proper time representation and utilizes the first way (109) of factorizing the square of the 3-derivative spinor operator (108). Surprisingly, a naive application of this alternative approach leads precisely to the value of $C_{T,6}$ in (16) corresponding to $\xi_{{}_{{\rm BT}}}$ in (6).

It is possible to generalize the $d=4$ (113) and $d=6$ (116) expressions for $C_{T}$ of the 3-derivative conformal fermion to any dimension $d$ obtaining the following counterpart of the general $d$ expressions for $C_{T}$ of the standard scalar and spinor (13), 4-derivative scalar (66) and 6-derivative scalar (75)

[TABLE]

It would be interesting to reproduce (118) in alternative flat-space approaches like the ones used in the higher-derivative scalar cases in Guerrieri:2016whh and Osborn:2016bev .

6 Conformal anomaly of 6d

higher derivative vector multiplet from

Seeley - DeWitt coefficient

Let us now rederive the above results for the ${\rm c}_{3}$ coefficient in (1) for the fields of the vector multiplet (8) by the same direct method as used in Bastianelli:2000hi to compute the conformal anomalies of the standard fields in the (2,0) tensor multiplet – using the general expression Gilkey:1975iq for the $b_{6}=\langle T^{\mu}_{\mu}\rangle$ Seeley-DeWitt coefficient of the 2nd order Laplace-type operator $\Delta=-D^{2}+X$.

The two key observations that allow one to do this are:252525We are grateful to D. Diaz for suggesting this approach to us. (i) like the higher derivative conformal scalar operators Gover:2005mn , the 4-derivative vector operator in (82) and the square of the 3-derivative spinor operator in (106) factorize into a product of 2nd order Laplacians on an Einstein space $R_{\mu\nu}={1\over 6}Rg_{\mu\nu}$ and thus their anomalies can be readily computed; (ii) considering a general Einstein background is sufficient to determine all the 4 anomaly coefficients ${\rm a},\ {\rm c}_{i}$ in (1). The special cases were already considered before – the 6-sphere (allowing to find the ${\rm a}$-coefficient Beccaria:2015uta ) and the Ricci-flat space (allowing to fix the ${\rm c}_{i}$ up to one free parameter Beccaria:2015ypa ). On a general Einstein background one may have both the scalar curvature and the Weyl tensor non-zero so that one may capture the $RC^{\mu\nu\lambda\rho}C_{\mu\nu\lambda\rho}$ terms in the expression for $b_{6}$ and thus determine one more combination of the Weyl anomaly coefficients.

As a result, in addition to the anomaly coefficients for the conformally coupled 2-derivative 6d scalar ($\Delta=-D^{2}+{1\over 5}R$) obtained in Bastianelli:2000hi

[TABLE]

we find for the 4-derivative conformal vector (82)

[TABLE]

and 3-derivative conformal MW spinor

[TABLE]

As a result, the anomaly coefficients for the higher derivative vector multiplet (8) are found to be262626This corrects the expressions for ${\rm c}_{i}$ in eq. (2.3) in Beccaria:2015ypa that assumed the wrong value of $\xi$ in (6).

[TABLE]

The values of ${\rm a}$-coefficient were found already in the special case of $S^{6}$ background in Beccaria:2015uta . The (1,0) supersymmetry constraint ${\rm c}_{1}-2{\rm c}_{2}+6{\rm c}_{3}=0$ in (2) and the relation ${\rm c}_{1}+4{\rm c}_{2}={62\over 45}$ were obtained by considering the Ricci-flat background in Beccaria:2015ypa . The coefficients ${\rm c}_{3}$ in (120) and (121) (or $C_{T,6}=3024{\rm c}_{3}$ in (11)) are the same as the ones found above in (92) and (116) from the computation of free energy on $S^{1}\times{\mathbb{H}}^{5}$. The values of ${\rm c}_{i}$ in (122) thus agree with (10) for the value of $\xi=-{8\over 9}$ found in Yankielowicz:2017xkf providing another independent confirmation of (7).

In Appendix C we shall present the extension of the computation presented in this section to more general $(1,0)$ superconformal multiplets with maximal spin 1 with the results that are again in agreement with (1) with (7).

Below we shall follow the notation in Bastianelli:2000hi and use that for an Einstein background one has $D_{\mu}R=0$ and $D^{\mu}C_{\mu\nu\lambda\rho}=0$ so that many terms in the general expression in $b_{6}$ simplify. The $E_{6}$ and $I_{1,2,3}$ invariants in (1) defined in Bastianelli:2000hi take the form272727 The only non zero total derivative terms among $C_{1,\dots,7}$ in Bastianelli:2000hi here are

$\displaystyle C_{5}$ $\displaystyle=C^{\alpha\beta\gamma\delta}D^{2}C_{\alpha\beta\gamma\delta}+(D_{\mu}C_{\alpha\beta\gamma\delta})^{2}=D_{\mu}(C_{\alpha\beta\gamma\delta}D^{\mu}C^{\alpha\beta\gamma\delta}),$

$\displaystyle C_{7}$ $\displaystyle=\tfrac{1}{12}RC_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}-\ C_{\alpha}{}^{\mu}{}_{\gamma}{}^{\nu}C^{\alpha\beta\gamma\delta}\ C_{\beta\mu\delta\nu}-\tfrac{1}{4}C_{\alpha\beta}{}^{\mu\nu}\ C^{\alpha\beta\gamma\delta}C_{\gamma\delta\mu\nu}+\tfrac{1}{4}(D_{\mu}C_{\alpha\beta\gamma\delta})^{2}.$

[TABLE]

Given a general scalar Laplacian

[TABLE]

the corresponding $b_{6}$ coefficient computed as in Bastianelli:2000hi is found to be ($\bar{b}_{6}\equiv(4\pi)^{3}b_{6}$)

[TABLE]

Expressing this in terms of the invariants in (1) using (123) and ignoring the total derivative terms we find that in the special case of the conformally coupled scalar when $\kappa={1\over 4}d(d-2)=6$ we reproduce the coefficients in (119).

The 4-derivative vector operator in (82) restricted to an Einstein background factorizes in the same way as in the sphere case discussed in Beccaria:2015uta : the action depends only on the transverse part $V^{\perp}_{\mu}$ of the vector and reduces to the integral of $V^{\perp}\Delta_{1\perp}(7)\,\Delta_{1\perp}(5)V^{\perp}$. The resulting partition function is then given by

[TABLE]

where like in (124) we defined $\Delta_{1}(\kappa)V_{\mu}\equiv(-D^{2}+\kappa\bar{R})V_{\mu}$ and $\Delta_{1\perp}$ is $\Delta_{1}$ restricted to $V^{\perp}_{\mu}$. The standard vector Laplacian on an Einstein background is $(-D^{2}g_{\mu\nu}+R_{\mu\nu})V^{\nu}=\Delta_{1}(5)V_{\mu}$. In (126) we used that $\det\Delta_{1}(\kappa)=\det\Delta_{1\perp}(\kappa)\,\det\Delta_{0}(\kappa-5)$.282828If $V_{\mu}=V^{\perp}_{\mu}+\partial_{\mu}\varphi$, then on an Einstein background one has (dropping total derivatives) $V^{\mu}\Delta_{1}(\kappa)V_{\mu}=V^{\mu\perp}\Delta_{1\perp}(\kappa)V^{\perp}_{\mu}+\varphi\Delta_{0}(\kappa-5)\varphi$. To find the $b_{6}$ coefficient for the vector Laplacian $\Delta_{1}(\kappa)$ from the general expressions in Gilkey:1975iq ; Bastianelli:2000hi one is to note that here the covariant derivative contains the extra vector connection part with the "internal" curvature $({\cal F}_{ij})_{\alpha}^{\ \ \beta}=R\indices{{}_{ij\alpha}^{\beta}}$. The analog of (6) then reads

[TABLE]

This generalizes the expression found in Bastianelli:2000hi in the special case of the standard vector Laplacian (corresponding to $\kappa=5$).

Eqs. (6) and (6) are all we need to compute the anomalies of the 4-derivative conformal vector since according to (126)

[TABLE]

As a result, one finds the coefficients given in (120).

The 3-derivative conformal spinor operator (106) restricted to an Einstein background becomes $\mathcal{O}^{(3)}=\not{D}^{3}+\tfrac{1}{30}R\not{D}$ so that its square factorizes as

[TABLE]

where $\Delta_{1\over 2}(\kappa)\equiv-D^{2}+\kappa\bar{R}$ acting on spinors has $D$ being spinor covariant derivative with the corresponding "internal" curvature ${\cal F}_{ij}=\tfrac{1}{4}\,R_{ijab}\,\gamma^{ab}$. The counterpart of (6) and (6) in the spinor case is then found to be292929Here we included the -1 fermion sign factor. $n_{f}$ is the complex dimension of the spinor space as in (13) equal to 2 in the 6d MW spinor case.

[TABLE]

For the standard spinor field with the squared operator being $\Delta_{1\over 2}(\kappa)$ with $\kappa={15\over 2}$ eq. (6) reproduces the coefficients in (1) found in Bastianelli:2000hi , i.e. for MW spinor with $n_{f}=2$ we get {\rm a}=-{191\over 288\times 7!}\ ,\ \ \ {\rm c}_{1}=-{224\over 3\times 7!},\ \ \ {\rm c}_{2}=-{8\over 7!},\ \ \ {\rm c}_{3}={10\over 7!}\. Using that for the 3-derivative spinor we have from (129) b_{6}(\psi^{(3)})=b_{6}\big{[}\Delta_{1\over 2}(\tfrac{15}{2})\big{]}+b_{6}\big{[}\Delta_{1\over 2}(\tfrac{13}{2})\big{]}, we find that the corresponding conformal anomaly coefficients are given by (121).

Acknowledgments

We would like to thank D. Butter, C.-M. Chang, D. Diaz, S. Giombi, Y.-H. Lin, H. Osborn, A. Petkou, S. Solodukhin, S. Yankielowicz and Y. Zhou for useful discussions and comments. The work of AAT was supported by the ERC Advanced grant no. 290456, the STFC Consolidated grant ST/L00044X/1, by the Australian Research Council project No. DP140103925 and the Russian Science Foundation grant 14-42-00047 at Lebedev Institute. The research of AAT at KITP was also supported in part by the National Science Foundation under grant No. NSF PHY11-25915. AAT also thanks the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work.

Appendix A Alternative computational scheme for free energy and $C_{T}$

of 3-derivative conformal spinor field

Let us start with the standard fermion case and compute the corresponding free energy without using the proper time representation for the log factor in (94) and doing the sum over $n$ first and the integral over $\lambda$ last. The sum over $n$ requires a regularization prescription and we shall adopt the same one as used, e.g., in Klebanov:2011uf 303030This relation may be justified, e.g., by first taking the derivative over ${\rm m}$, then doing the convergent sum using $\sum^{\infty}_{n=-\infty}{1\over[(n+\gamma)^{2}+q^{2}{\rm m}^{2}]}={\pi\sinh(2\pi q\,{\rm m})\over q\,{\rm m}\,[\cosh(2\,\pi\,q\,{\rm m})-\cos(2\,\pi\,\gamma)]}$, and finally integrating back over ${\rm m}$ (assuming also that $\sum^{\infty}_{n=-\infty}c=0$). Note that the choice of integration constants or regularization involved may break the formal invariance under the integer shifts of $\gamma$.

[TABLE]

It is important to stress that because of the regularization involved this relation directly applies for $\gamma<1$ (and $q^{2}{\rm m}^{2}>-1$ if $\gamma=0$); the expressions found using (131) with parameters outside that range should be defined by an analytic continuation. The case of half-integer summation in (94) corresponds to the legitimate values $\gamma=\tfrac{1}{2}$, ${\rm m}^{2}=\lambda\geq 0$. As a result, we get

[TABLE]

The integral over $\lambda$ is divergent at large $\lambda$; omitting the power divergent part proportional to $q$ one reproduces the same $d=4$ and $d=6$ expressions as in (100) and (103). The second $q$-derivative of ${\cal F}_{q}$ is always finite and using (29), we then reproduce from (132) the standard result for $C_{T,d}(\psi)=\tfrac{1}{2}n_{f}\,d$  given in (13).

In the case of the 3-derivative spinor we may use the factorized expression (109) for the square of its kinetic operator leading to the following expression for the free energy that generalizes (94)

[TABLE]

Computing the sum in $K(\lambda,q)$ using the prescription (131) (with $\gamma$ equal to $\tfrac{1}{2},\ q+\tfrac{1}{2},\ q-\tfrac{1}{2}$) we find the following analog of (132)

[TABLE]

Taking the second derivative of (135) over $q$ at $q=1$ and computing the resulting finite integral over $\lambda$ we find, according to (29), the following expression for the $C_{T}$ coefficient for the 3-derivative conformal fermion in $d$ dimensions (with $n_{f}=2$ in $d=4$ and $d=6$)

[TABLE]

Remarkably, the $d=6$ value is precisely the one in (16) corresponding to $\xi_{{}_{{\rm BT}}}$ in (6). However, a warning sign is that the $d=4$ value disagrees with the correct one $C_{T,4}=-{4\over 3}$ in (113) corresponding to ${\rm c}=-{1\over 120}$ found by direct computation in Fradkin:1981jc ; Fradkin:1985am .

This suggests some problem with the above computation. Indeed, while the representation (132) for the free energy of the standard fermion is true for any $q$, the expression (135) that was obtained using (131) with $\gamma=q\pm\tfrac{1}{2}$ is formally valid only for $0\leq|q|<\tfrac{1}{2}$. It cannot thus be differentiated directly at $q=1$ and this is the reason why the resulting values of $C_{T}\sim{\cal F}^{\prime\prime}_{1}$ or (136) are not correct.

The correct procedure is to first evaluate (135) for $0\leq|q|<\tfrac{1}{2}$, then analytically extend the resulting expression for ${\cal F}_{q}$ to all values of $q$ and finally differentiate it over $q$ obtaining, in particular, the corresponding Rényi entropy (25) and $C_{T}\sim{\cal F}^{\prime\prime}_{1}$ in (29). The results will then agree with (112),(113) and (115),(116) found using the heat-kernel regularization approach used in the main text.

To see this explicitly let us note that the $+2\cos(2\,\pi\,q)=-2\cos[2\pi(q\pm\tfrac{1}{2})]$ term in (135) originated from the $\gamma=q\pm\tfrac{1}{2}$ shifts in (131). To make the use of (131) legitimate we may first formally replace this shift by $\gamma={q\over k}\pm\tfrac{1}{2}$, evaluate (135) for $k>2$ and then analytically continue $k\to 1$ in the final result. Replacing $\cos(2\,\pi\,q)\to\cos(2\,\pi\,{q\over k})$ in (135) we find after computing the second derivative of (135) at $q=1$ in $d=4,\ 6$ for $k>2$ (see (30))

[TABLE]

These expressions indeed reproduce the correct values $C_{T,4}=-{2\over 3}n_{f}$ in (113) and $C_{T,6}=-{18\over 5}n_{f}$ in (113) after the analytic continuation to $k=1$.313131Note that for $k\to\infty$ the expressions in (137) become 3 times the standard fermion values in (101) and (104). The reason for this is that in this limit the $\pm q\to\pm{q\over k}$ shifts of $n$ in (134) disappear and we get the 3rd power of the standard fermion expression under the log.

Appendix B 2-derivative non-gauge conformal vector

Here we shall follow Erdmenger:1997gy ; Osborn:2015rna and consider a non-unitary theory described by 2-derivative vector field with conformal but not gauge-invariant action for $d\not=4$.323232Partition function of a similar 2-derivative spin 2 theory was discussed in Beccaria:2015vaa . The corresponding Weyl-invariant curved space action is

[TABLE]

It is equivalent to the standard Maxwell action for $d=4$. The corresponding $C_{T}$ coefficient found in Osborn:2015rna is

[TABLE]

Specifying to the case of the $S^{1}\times\mathbb{H}^{d-1}$ background and separating $V_{0}$ and $V_{i}=V^{\perp}_{i}+\partial_{i}\,\varphi$ components as in (84) we find that the corresponding partition function has two contributions: one from ${V}^{\perp}_{i}$ and one corresponding to the $\partial^{4}$ conformal scalar in $d\neq 4$. The scalar part is absent in $d=4$ due to gauge invariance that is then present in (138) (cf. section 4.1 and Beccaria:2014jxa ).

From (138) we get the following mixed Lagrangian for $\chi\equiv V_{0},\ \varphi$ and $V^{\perp}_{i}$

[TABLE]

where

[TABLE]

Using that on ${S}^{1}\times\mathbb{H}^{d-1}$ we have $D_{i}D^{2}D_{i}\varphi=\partial_{0}^{2}\bm{D}^{2}+(\bm{D}^{2})^{2}-(d-2)\bm{D}^{2},$ we obtain for the scalar part of (140)

[TABLE]

Integrating over $\chi$ and $\varphi$ in the path integral, their contribution can be represented in terms of the determinant of the following 6-order scalar operator

[TABLE]

The determinant of the $\bm{D}^{2}$ factor cancels against the Jacobian of the change of variables $V_{i}\to V^{\perp}_{i}+\partial_{i}\,\varphi$, while the remaining 4-order scalar operator is equivalent to the conformal $\partial^{4}$ one which factorizes as in (57) with the eigenvalues given in (59).

As a result, we find that

[TABLE]

In view of the expression (66) for $C_{T,d}(\varphi^{(4)})$, to match (139) we should thus get

[TABLE]

The transverse spatial vector part of the free energy that follows from (141) is given by (see (37) for $s=1$, (38) and (131))333333Here we use (131) with $\gamma=0$ so the result is equivalent to the one in the heat kernel approach used in the main text.

[TABLE]

Computing the corresponding $C_{T}$ according to (29) we find

[TABLE]

in agreement with (145). This provides an alternative derivation of (139).

Appendix C Conformal anomalies of general higher derivative short

superconformal 6d vector multiplets

The calculation of the full 6d conformal anomaly of the $V^{(1,0)}$ multiplet from the Seeley-DeWitt coefficients on an Einstein background presented in section 6 may be generalized to other (1,0) superconformal vector multiplets. $V^{(1,0)}\equiv V^{(1,0)}_{p=2}$ is the lowest member of a family of multiplets $V^{(1,0)}_{p}$ ($p=2,3,4,\dots$) that contain scalars, spinors and vectors with $p$-dependent higher-derivative kinetic terms.

In terms of $OSp(2,6|2)$ representations Minwalla:1997ka ; Dobrev:2002dt ; Bhattacharya:2008zy the hypermultiplet $S^{(1,0)}$ is a doubleton ultra-short representation Ferrara:2000xg . New (possibly massive) conformal representations are obtained by tensoring $p$ copies of $S^{(1,0)}$. The resulting multiplets $V^{(1,0)}_{p}$ are short with the maximal spin equal to 1. The structure of these multiplets Gimon:1999yu is shown in Table 1.

Here $\Delta$ is the scaling dimension of the conformal group $SO(2,6)$ related to the canonical dimension of the corresponding 6d field $\Phi$ by $\dim\Phi=6-\Delta$. We indicated also the $SU(2)$ R-symmetry representations. $\psi^{\pm}$ are positive/negative chirality MW spinors while $\varphi$ and $\varphi^{\prime}$ are scalars. The vector $V_{\mu}$ has conformal but not gauge invariant action for $p>2$. From the canonical dimensions one can determine the number of derivatives in kinetic terms in the corresponding 6d Lagrangian that has the following schematic form

[TABLE]

where fields transform under $SU(2)_{R}$ according to Table 1.

The higher derivative operators in (148) should have a covariant and Weyl-invariant generalization to curved background. Remarkably, just as for the $p=2$ case discussed in section 6 all these operators for conformal fields in Table 1 factorize on an Einstein space background. Let us denote by $\Phi^{(n)}$ a field with $n$ derivatives in the kinetic term. For higher derivative conformal scalar $\varphi^{(2n)}$ operators (GJMS operators) the factorization on an Einstein space reads Gover:2005mn

[TABLE]

where $\Delta_{0}(\kappa)$ is the scalar Laplacian defined in (124). Similarly, for the $(2n+1)$-derivative spinor $\psi^{(2\,n+1)}$, the square of the corresponding conformal operator factorizes as fischmann2015conformal

[TABLE]

which generalizes the $n=1$ expression in (129). Finally, for the vectors $V^{(2n)}$, with $n>2$, the factorization on a general Einstein space looks the same as on the 6-sphere and can be found from (A.17) of Beccaria:2015uta for the massive conformal representation $[\Delta,\bm{h}]=[3+n,(1,0,0)]$. The corresponding partition function can be written as (cf. (126))

[TABLE]

These factorizations into 2nd order Laplacians allow us to compute the corresponding conformal anomalies using the same method as in the $V^{(1,0)}$ (i.e. $p=2$) case in section 6. Using the expressions for the Seeley - DeWitt coefficients $b_{6}$ in (6),(6),(6) we can compute the anomalies of the fields in the $V_{p}^{(1,0)}$ multiplet with the results summarized below343434Our discussion is formal as curved-space higher-derivative operators in a given dimension (here $d=6$) may exist only to some critical order (as is well known in the scalar GJMS case).

[TABLE]

The total results for the multiplet $V_{p}^{(1,0)}$ are then

[TABLE]

The expressions in (153) are in perfect agreement with (3),(2),(1) with (7) as one can see using that for general $p$ the coefficients in the anomaly polynomial are Beccaria:2015ypa

[TABLE]

Remarkably, these expressions continue to hold also for $p=2$ where (151) should be replaced by (126).353535This agreement is not accidental. For $p=2$ the vector $V^{(4)}$ has gauge invariant action (it is described by the conformal representation $[5,(1,0,0)]-[6,(0,0,0)]$ where the subtraction takes into account the gauge invariance), while the scalar $\varphi^{\prime}$ is absent in the multiplet $V^{(1,0)}$. In the general expression for the partition function for the multiplet $V^{(1,0)}_{p}$ continued formally to $p=2$ the field $\varphi^{\prime}$ enters effectively with a negative multiplicity and thus contributes precisely like the ghost scalar factor in (126).