# Casimir energy on the sphere and 6D CFT trace anomaly

R Aros1 F Bugini2 DE Díaz3 and C Núñez-Barra4 1 Departamento de Ciencias
Fisicas, Universidad Andres Bello, Sazie 2212, Piso 7, Santiago, Chile  2
Departamento de Matemática y Física Aplicadas, Universidad Católica de la
Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile  3 Departamento
de Ciencias Fisicas, Universidad Andres Bello, Autopista Concepcion-Talcahuano
7100, Talcahuano, Chile  4 Facultad de Física, Pontificia Universidad Católica
de Chile, Vicuña Mackenna 4860, Santiago, Chile
raros@unab.cl,fbugini@ucsc.cl,danilodiaz<EMAIL_ADDRESS>
###### Abstract

We elucidate the dependence of the Casimir energy on the trace anomaly
coefficients for a six-dimensional CFT on $R\times S^{5}$. This extends the
universal dependence on the central charge in 2D and the relation by Cappelli
and Coste in 4D, unveiling the role of the trivial total derivatives in the
anomaly that renders the Casimir energy scheme dependent. We obtain

$E_{o}=-\frac{15}{8}\,a_{6}-\frac{5}{12}\,\left(g_{5}+\frac{1}{4}\,g_{7}+\frac{1}{2}\,g_{8}-10\,g_{9}+g_{10}\right),$

with $a_{6}$ being the type A central charge and the $g$’s, the coefficients
of five out of six terms that form a basis for trivial total derivatives. The
derivation is based on the Polyakov formulas (conformal primitive) resulting
from the integration of the trace anomaly.
Alternatively, on a 6D conformally flat background the above basis is
redundant and one can simplify further to get, in terms of the Schouten scalar
$J$ and the Schouten tensor $V$, Branson’s basis for trivial total derivatives
$\nabla^{2}\nabla^{2}J$, $\nabla^{2}J^{2}$ and
$\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla\,J)$ with coefficients
$\gamma_{1},\gamma_{2}$ and $\gamma_{3}$, respectively,

$E_{o}=-\frac{15}{8}a_{6}-\frac{1}{24}\left(\gamma_{1}-\gamma_{2}-\frac{1}{8}\gamma_{3}\right)~{}.$

##  1 Introduction

The Casimir effect [1] is a remarkable macroscopic quantum response of a
system to external influences (e.g. boundary conditions or background fields).
It has been extensively studied over the years and experimentally verified [2,
3] (for a recent review, see [4]). One interesting route contemplates the
extension of the original setting regarding the 4D electromagnetic field to a
conformal field theory in a generic even dimension. Since conformal symmetry
is quite restrictive, the Casimir energy of a 2D CFT on the cylinder is
universal and given by the ubiquitous central charge $c$ [5, 6] of the trace
anomaly,

$E_{o}=-\frac{c}{12}.$ (1)

The aforementioned universality is partially lost in 4D, yet the trace anomaly
coefficients determine the vacuum energy111We stick to the conventions of [8].
as shown by Cappelli and Coste [7]

$E_{o}=\frac{3}{4}\,a+\frac{3}{8}\,g~{},$ (2)

where $a$ is the universal type A trace anomaly coefficient and $g$ is the
regularization scheme dependent coefficient of the total derivative term

$(4\pi)^{2}\langle T\rangle=-a\,E_{4}+c\,W^{2}+g\,\nabla^{2}R~{}.$ (3)

The total derivative term $\nabla^{2}R$ is a trivial anomaly since it comes
from the conformal variation 222By conformal we mean a local Weyl rescaling of
the metric. We will use both terms, conformal and Weyl, indistinctly in what
follows. of a conformal primitive that happens to be a local curvature
invariant with the appropriate scaling dimension333The other two candidates
for a local curvature invariant $Riem^{2}$ and $Ric^{2}$ can be reduced to
$R^{2}$ by exploiting the conformal invariance of the Euler density $E_{4}$
and of the quadratic contraction of the Weyl tensor $W^{2}$., namely $R^{2}$,
that can simply be added to the action.

There are good indications that in higher (necessarily even) dimensions the
Casimir energy on the cylinder 444Meaning that the spatial section is
compactified to a sphere. is still dictated by the trace anomaly of the CFT.
The structure of the latter was asserted years ago by Deser and Schwimmer [9]
to consist of 555See also [10, 11, 12] for a cohomological approach. For the
closely related question of the classification of global conformal invariants,
see [13] on conformally flat manifolds and [14, 15] in greatest generality. a
multiple of the Euler density (type A), a point-wise Weyl invariant (type B)
and a trivial total derivative

$\displaystyle(4\pi)^{\frac{n}{2}}\cdot\langle T\rangle$ $\displaystyle=$
$\displaystyle-(-)^{\frac{n}{2}}\,a_{n}\cdot E_{n}+\left\\{\mbox{Point-wise
Weyl Invariant}\right\\}+\mbox{t.t.d.}$

Remarkably, a simple relation between the Casimir energy on the cylinder and
the type A trace anomaly coefficient was found by Herzog and Huang in [16]

$E_{o}=\frac{(-1)^{\frac{n}{2}}\cdot(n-1)!!}{2^{\frac{n}{2}}}\cdot a_{n}~{},$
(4)

but only valid in a particular regularization scheme where the trivial total
derivatives are absent. For example, in 6D one encounters discrepancies with
the Casimir energy for the free (2,0) tensor multiplet as computed via
standard $\zeta$-function regularization (see, e.g. [17, 18]). A 6D relation,
like the Cappelli and Coste one, taking into account the role of the trivial
total derivatives in the trace anomaly, is clearly still missing. This note
aims to provide such an extension.

Our results are based on the following rather simple observation: since the
trivial total derivatives in the trace anomaly stem from a local conformal
invariant in the action, it is enough to evaluate the latter on the geometry
of the cylinder to obtain its contribution to the partition function and, in
consequence, to the vacuum energy. For completeness, we will also re-derive
the dependence on the type A central charge by the direct examination of the
Polyakov formulas that provide the conformal primitive of the trace anomaly.
We prefer Branson’s construction where the Euler density is traded in for the
Q-curvature, this reduces the answer to terms at most quadratic in the Weyl
factor that relates the standard metric on the cylinder to that of a flat
shell. Although our derivation is conceptually clear, in practice we face the
complication that, unlike the Euler term, the number of independent trivial
total derivatives increases significantly with the dimension; there are none
in 2D, one in 4D, six in 6D, whereas 8D and beyond remain largely unexplored
666There is, though, a basis of nine local conformal primitives on conformally
flat backgrounds in 8D due to Branson and Peterson [19]..

We start in Section 2 with a brief account of Polyakov formulas. We then
consider the explicit cases of 2D, 4D, and 6D in Section 3 and test in Section
4 against data available in the literature. We then conclude and collect some
useful relations in the appendix.

##  2 Polyakov formulas, Q-curvature, and local curvature invariants

The (generalized) Polyakov formula provides a conformal primitive of the trace
anomaly, as first shown by Polyakov in 2D [20]. In higher even dimensions,
Branson asserted the form of the corresponding extension by making use of the
suitable features of the critical Q-curvature (see, e.g. [21]). In a (closed
and with Euclidean signature) conformally flat manifold the topological
content of the trace anomaly is also captured by the Q-curvature and all
point-wise Weyl invariants vanish so that the structure of the trace anomaly
is significantly simplified

$\displaystyle(4\pi)^{\frac{n}{2}}\cdot\langle T\rangle$ $\displaystyle=$
$\displaystyle-(-)^{\frac{n}{2}}\,n\cdot a_{n}\cdot Q_{n}+\mbox{t.t.d.}$ (5)

Here one exploits the linear dependence of the variation of the Q-curvature on
the Weyl factor $\sigma$, that conformally relates two metrics
$\hat{g}=e^{2\,\sigma}g$, and integrates in an auxiliary parameter $t$ along
any interpolating path $\sigma(t)$ such that $\sigma(0)=0$ and
$\sigma(1)=\sigma$, to obtain for the partition functions

$\displaystyle\Gamma[g,\hat{g}]=-\Gamma[\hat{g},g]$ $\displaystyle=$
$\displaystyle\log Z[\hat{g}]-\log Z[g]=-(-)^{\frac{n}{2}}\,\frac{n\cdot
a_{n}}{(4\pi)^{\frac{n}{2}}}\int_{M_{n}}dv_{g}\;\sigma\cdot\left\\{Q_{n}+\frac{1}{2}\,P_{n}\,\sigma\right\\}$
(6)
$\displaystyle+\left\\{\int_{\hat{M}_{n}}dv_{\hat{g}}\;\hat{\mathcal{F}}-\int_{M_{n}}dv_{g}\;{\mathcal{F}}\right\\}~{}.$

The explicit dependence on $\sigma$ is at most quadratic and the GJMS operator
$P_{n}$ can be integrated by parts to yield the so-called dilaton action for
the Weyl factor, whereas higher order terms in $\sigma$ are hidden in the
finite conformal variation of the local conformal primitive $\mathcal{F}$.

Now, considering the standard metric on the cylinder $R\times S^{n-1}$
conformally related to flat space

$\hat{g}=du^{2}+d\Omega^{2}=r^{-2}\left\\{dr^{2}+r^{2}\,d\Omega^{2}\right\\}$
(7)

via a Weyl factor $\sigma=-\ln r$, we can readily read off the Casimir energy
resulting from the compactification of the spatial section referred to the
vanishing vacuum energy of flat space. Alternatively, one can consider a
temperature circle with inverse temperature $\beta$ and read off the dominant
vacuum term as $\beta$ goes to infinity on $S^{1}_{\beta}\times S^{n-1}$.
In “flatland” (meaning flat space or an spherical shell for finite inverse
temperature), the only contribution comes from the nth-order kinetic term
$|\partial\partial^{2}...\partial^{2}\sigma|^{2}$ or
$|\partial^{2}...\partial^{2}\sigma|^{2}$ depending on the (even) dimension
$n$, that produces the same answer $((n-2)!!)^{2}/r^{n}$ so that the radial
integral results in $\int\frac{dr}{r}=\int du$ or $\beta$. Performing the
remaining volume integral on the sphere, the generic relation for the Casimir
energy (modulo trivial total derivative contributions) becomes

$E_{o}=\frac{(-1)^{\frac{n}{2}}\cdot n!!}{2^{\frac{n}{2}+1}}\cdot
a_{n}+...~{}.$ (8)

This ought to be compared with the result by Herzog and Huang in the Euler
basis

$E_{o}=\frac{(-1)^{\frac{n}{2}}\cdot(n-1)!!}{2^{\frac{n}{2}}}\cdot
a_{n}+...~{}.$ (9)

The apparent discrepancy above stems from the fact that each result is valid
within a different regularization scheme where trivial total derivatives are
discarded. However, evaluation of the local conformal primitive of the trivial
total derivatives on the cylinder will complement the result and the agreement
between the two schemes and a more general one will be achieved. The local
conformal primitive vanishes on flat space, and must be a constant on the
cylinder. This yields the scheme-dependent contribution to the Casimir energy
foreseen above. The only technical difficulty in carrying out the above
completion, as already pointed out, lies in the increasing complexity of the
trivial total derivative terms and their conformal primitives with the
dimension.

##  3 Casimir energy and trace anomaly: the explicit relation

In what follows we will revisit the 2D and 4D cases and obtain, for the first
time to our knowledge, the 6D extension of the Capelli-Coste relation.

###  3.1 2D CFT

Let us start by reproducing the 2D relation 777This was essentially obtained
by Affleck [6], except for the fact that his result applies to the modular
transformed torus, i.e., to the Cardy or high temperature regime. following
Cappelli and Coste. There are no trivial anomalies in 2D, by convention
$Q_{2}=\frac{R}{2}$ is the Gaussian curvature of the closed surface and
$P_{2}=-\nabla^{2}$, that is integrated by parts to write down the answer in
the well-known Liouville form

$\Gamma[g,\hat{g}]=\frac{a_{2}}{4\pi}\int_{\text{flatland}}dv_{g}\;\left\\{\;|\nabla\sigma|^{2}+\sigma\cdot
R\right\\}$ (10)

From this potential for the Weyl factor, also known in the conformal geometry
literature as the Moser functional, evaluated on the fiducial flat metric with
$R=0$ and for the conformal factor $|\partial\sigma|^{2}=\frac{1}{r^{2}}$, one
can readily obtain the leading contribution to the partition function as
$\beta\rightarrow\infty$

$-\beta\cdot E_{o}=\frac{a_{2}}{4\pi}\cdot
2\pi\cdot\beta=\frac{a_{2}}{2}\cdot\beta$ (11)

This corresponds to the more familiar $E_{o}=-c/12$, when the central charge
is tuned so that for the scalar Laplacian $a_{2}=1/6$ one gets $c=1$, i.e.,
$a_{2}=c/6$. In all,

$E_{o}=-\frac{a_{2}}{2}.$ (12)

###  3.2 4D CFT

Now we revisit the derivation by Cappelli and Coste in 4D, but in the light of
Branson’s insight that exploited the suitable transformation law of the
Q-curvature under Weyl rescaling of the metric (see also Riegert’s action [22]
in local form). The trace anomaly at a conformally flat background becomes

$(4\pi)^{2}\cdot\langle T\rangle=-a_{4}\cdot
E_{4}+g\cdot\nabla^{2}\,R=-4\,a_{4}\cdot
Q_{4}+\tilde{g}\cdot\nabla^{2}\,R~{},$ (13)

and its conformal primitive, referred to the fiducial flat metric, is then
given by

$\displaystyle\Gamma[g,\hat{g}]$ $\displaystyle=$
$\displaystyle-\frac{a_{4}}{4\pi^{2}}\int_{\text{flatland}}dv_{g}\left\\{\sigma\cdot
Q_{4}+\frac{1}{2}\,\sigma\cdot
P_{4}\,\sigma\right\\}+\frac{\tilde{g}}{16\pi^{2}}\int_{S^{1}_{\beta}\times
S^{3}}dv_{\hat{g}}\;\frac{-1}{12}\hat{R}^{2}$ (14) $\displaystyle=$
$\displaystyle-\frac{a_{4}}{8\pi^{2}}\int_{\text{flatland}}dv_{g}\;|\partial^{2}\sigma|^{2}+\frac{\tilde{g}}{16\pi^{2}}\int_{S^{1}_{\beta}\times
S^{3}}dv_{\hat{g}}\;\frac{-1}{12}\hat{R}^{2}~{}.$

We obtain contribution from the fourth-order kinetic term
$|\partial^{2}\sigma|^{2}=\frac{4}{r^{4}}$ and from the local curvature
invariant $\frac{-1}{12}\hat{R}^{2}=-3$, conformal primitive of the trivial
total derivative, so that we end up with

$-\beta\cdot
E_{o}=-\left\\{\frac{a_{4}}{2\pi^{2}}+\frac{3\tilde{g}}{16\pi^{2}}\right\\}\cdot
2\pi^{2}\cdot\beta=-\left\\{a_{4}+\frac{3}{8}\,\tilde{g}\right\\}\cdot\beta~{}.$
(15)

Therefore, we obtain for the Casimir energy

$E_{o}=a_{4}+\frac{3}{8}\,\tilde{g}~{}.$ (16)

To translate to the Euler basis, it is enough to remember that on conformally
flat backgrounds $4Q_{4}=E_{4}-\frac{2}{3}\nabla^{2}R$, so that
$\tilde{g}=g-\frac{2}{3}a_{4}$ and finally obtain the original result by
Cappelli and Coste888Our convention follows [8] and slightly differs from that
of the original Cappelli-Coste paper [7].

$E_{o}=\frac{3}{4}\,\left(a_{4}+\frac{1}{2}\,g\right)~{}.$ (17)

Notice that the precise shift $\tilde{g}=g-\frac{2}{3}a_{4}$ decorates the
dilaton action (eq. 14), by using the conformal transformation of
$\sqrt{\hat{g}}\hat{R}^{2}$, as follows

$\displaystyle\Gamma[\delta,\hat{\delta}]$ $\displaystyle=$
$\displaystyle-\frac{a_{4}}{8\pi^{2}}\int_{\text{flatland}}dv_{\delta}\;|\partial^{2}\sigma|^{2}-\frac{2}{3}\frac{a_{4}}{16\pi^{2}}\int_{\text{flatland}}dv_{\delta}\;\frac{-1}{12}\left\\{-6\partial^{2}\sigma-6|\partial\sigma|^{2}\right\\}^{2}\,+...$
(18) $\displaystyle=$
$\displaystyle\frac{a_{4}}{8\pi^{2}}\int_{\text{flatland}}dv_{\delta}\;\left\\{2|\partial\sigma|^{2}\partial^{2}\sigma+(|\partial\sigma|^{2})^{2}\right\\}\,...~{},$

where the ellipsis stands for the contribution of the trivial anomaly
$g\nabla^{2}R$. Evaluating the first part on the cylinder will again produce
the dependence of the vacuum energy on the central charge $a$, while the
contribution from the trivial anomaly can easily be reintroduced by evaluating
the associated counterterm (conformal primitive) 999 This shortcut route to
the Cappelli-Coste relation was somehow already implicit in [23](see appendix
A.1 therein)..

###  3.3 6D CFT

It is at 6D that a subtle departure between total derivatives and trivial
total derivatives occurs, stemming out from the difference between global
conformal invariants and the trace anomaly of a CFT. The latter must comply
with the Wess-Zumino consistency or integrability condition, requiring that
the allowed total derivatives must be derived from the conformal variation of
a local curvature invariant. In 6D, a basis for total derivatives is provided
by the seven divergences $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$ and $C_{7}$ of
[24], however once they are expressed in the reduced basis for trivial total
derivatives of $M_{5},M_{6},M_{7},M_{8},M_{9}$ and $M_{10}$ there remains an
“offending term” $\nabla^{2}|Ric|^{2}$. The basis for trivial total
derivatives was obtained in [25] by performing the conformal variation of all
possible curvature invariants with the appropriate scaling dimension. There
are seventeen elements $K_{1},...,K_{17}$, from which seven combinations are
the total derivatives $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6},C_{7}$ whose
integrals vanish identically, another combination gives the Euler density
$E_{6}$ which is a topological invariant, and there are also three other
combinations that produce three independent point-wise conformal invariants
$I_{1},I_{2}$ and $I_{3}$, so that in all there remain six independent
combinations -conformal primitives that we denote
$F_{5},F_{6},F_{7},F_{8},F_{9},F_{10}$ \- whose conformal variations produce
the trivial total derivatives $M_{5},M_{6},M_{7},M_{8},M_{9},M_{10}$ (we
collect explicit expressions in A).

Therefore, the 6D CFT trace anomaly at a conformally flat background becomes

$\displaystyle(4\pi)^{3}\cdot\langle T\rangle$ $\displaystyle=$ $\displaystyle
a_{6}\cdot E_{6}+g_{5}\cdot M_{5}+g_{6}\cdot M_{6}+g_{7}\cdot M_{7}+g_{8}\cdot
M_{8}+g_{9}\cdot M_{9}+g_{10}\cdot M_{10}$ (19) $\displaystyle=$
$\displaystyle 6\,a_{6}\cdot Q_{6}+\tilde{g}_{5}\cdot M_{5}+\tilde{g}_{6}\cdot
M_{6}+\tilde{g}_{7}\cdot M_{7}+\tilde{g}_{8}\cdot M_{8}+\tilde{g}_{9}\cdot
M_{9}+\tilde{g}_{10}\cdot M_{10}~{}.$

The conformal primitive, referred to a fiducial flat metric where all
curvature invariants vanish, is then given by

$\displaystyle\Gamma[g,\hat{g}]$ $\displaystyle=$
$\displaystyle\frac{3\,a_{6}}{64\pi^{3}}\int_{\text{flatland}}dv_{g}\;|\partial\,\partial^{2}\sigma|^{2}$
$\displaystyle+\frac{1}{64\pi^{3}}\int_{S^{1}_{\beta}\times
S^{5}}dv_{g}\left\\{\tilde{g}_{5}\cdot\mathcal{\hat{F}}_{5}+{\tilde{g}_{6}}\cdot\mathcal{\hat{F}}_{6}+{\tilde{g}_{7}}\cdot\mathcal{\hat{F}}_{7}+{\tilde{g}_{8}}\cdot\mathcal{\hat{F}}_{8}+{\tilde{g}_{9}}\cdot\mathcal{\hat{F}}_{9}+{\tilde{g}_{10}}\cdot\mathcal{\hat{F}}_{10}\right\\}.$

When evaluated at $S^{1}_{\beta}\times S^{5}$ with the standard product
metric, we obtain contributions from the sixth-order kinetic term
$|\nabla\,\nabla^{2}\sigma|^{2}=\frac{64}{r^{6}}$ and from the local curvature
invariants

$\displaystyle\mathcal{\hat{F}}_{5}=\frac{80}{3}\quad,\quad\mathcal{\hat{F}}_{6}=0\quad,\quad\mathcal{\hat{F}}_{7}=\frac{20}{3}\quad,$
$\displaystyle\mathcal{\hat{F}}_{8}=\frac{40}{3}\quad,\quad\mathcal{\hat{F}}_{9}=-\frac{800}{3}\quad,\quad\mathcal{\hat{F}}_{10}=\frac{80}{3}\quad.$
(21)

Collecting all contributions

$\displaystyle-\beta\cdot E_{o}$ $\displaystyle=$
$\displaystyle\frac{1}{64\pi^{3}}\left\\{3a_{6}\cdot
64+\tilde{g}_{5}\cdot\frac{80}{3}+\tilde{g}_{7}\cdot\frac{20}{3}+\tilde{g}_{8}\cdot\frac{40}{3}+\tilde{g}_{9}\cdot\frac{-800}{3}+\tilde{g}_{10}\cdot\frac{80}{3}\right\\}\cdot\pi^{3}\cdot\beta$
(22) $\displaystyle=$
$\displaystyle\left\\{3a_{6}+\frac{5}{12}\,\left[\tilde{g}_{5}+\frac{1}{4}\,\tilde{g}_{7}+\frac{1}{2}\,\tilde{g}_{8}-10\,\tilde{g}_{9}+\tilde{g}_{10}\right]\right\\}\cdot\beta~{}.$

Therefore, we obtain for the Casimir energy

$E_{o}=-3\,a_{6}-\frac{5}{12}\,\left[\tilde{g}_{5}+\frac{1}{4}\,\tilde{g}_{7}+\frac{1}{2}\,\tilde{g}_{8}-10\,\tilde{g}_{9}+\tilde{g}_{10}\right]~{}.$
(23)

To translate to the Euler basis we have to express the 6D Q-curvature in terms
of the Euler density and trivial total derivatives in a conformally flat
background. The linearity under conformal transformations, a defining property
of the Q-curvature, is shared by the whole family of pondered Euler densities
$\tilde{E}_{6}$ introduced by Anselmi in [26] (see [25] as well). In our
conventions, the dictionary goes as follows

$E_{6}=\tilde{E}_{6}+\left(\frac{67}{10}-\frac{5}{2}\zeta\right)M_{6}+\left(\frac{5}{2}\zeta-\frac{51}{5}\right)M_{7}+\left(\frac{1}{16}\zeta-\frac{9}{200}\right)M_{9}-\frac{3}{5}M_{10}~{},$
(24)

and results in the shift of the trivial anomaly coefficients

$\displaystyle\tilde{g}_{5}=g_{5}\quad$ ,
$\displaystyle\quad\tilde{g}_{6}=g_{6}+a_{6}\left(\frac{67}{10}-\frac{5}{2}\zeta\right)\quad,\quad\tilde{g}_{7}=g_{7}+a_{6}\left(\frac{5}{2}\zeta-\frac{51}{5}\right)\quad,$
$\displaystyle\tilde{g}_{8}=g_{8}\quad$ ,
$\displaystyle\quad\tilde{g}_{9}=g_{9}+a_{6}\left(\frac{1}{16}\zeta-\frac{9}{200}\right)\quad,\quad\tilde{g}_{10}=g_{10}-\frac{3}{5}a_{6}~{}.$
(25)

Substituting back in the Casimir energy relation (eqn.23), we obtain the
alternative relation that completes the partial result obtained by Herzog and
Huang in 6D

$E_{o}=-\frac{15}{8}\,a_{6}-\frac{5}{12}\,\left[{g}_{5}+\frac{1}{4}\,{g}_{7}+\frac{1}{2}\,{g}_{8}-10\,{g}_{9}+{g}_{10}\right]~{}.$
(26)

The above relation is robust in the sense that any construct with the same
conformal transformation property as the critical Q-curvature $Q_{6}$ will
produce the same result. For completeness, let us also examine the two-
parameter family of trivial total derivative terms that respects the linearity
in a generic background 101010We are grateful to I.L. Shapiro for a useful
discussion on this point. found out by Hamada [27] (eqn.4.20 and 4.21 therein,
see also [28, 29]) at the expense of modifying Branson’s operator $P_{6}$ :

$\displaystyle(i)$
$\displaystyle\frac{3}{4}\nabla^{2}W^{2}-3\nabla\nabla(WW)\,=\,-\frac{3}{4}C_{5}+3C_{7}\,=\,\frac{1}{4}M_{5}+\frac{1}{2}M_{6}-\frac{3}{4}M_{7}+M_{8}+\frac{9}{160}M_{9}$
(27) $\displaystyle(ii)$
$\displaystyle\frac{3}{2}\nabla^{2}W^{2}\,=\,3C_{5}\,=\,-3M_{6}+3M_{8}+\frac{3}{20}M_{9}$
(28)

It is reassuring to verify that none of the two combinations,
$\nabla\nabla(WW)$ nor $\nabla^{2}W^{2}$, will produce a non-vanishing result
in the relation for the Casimir energy. The above freedom agrees with the
usual wisdom in Conformal Geometry that the pair of critical Q-curvature
$Q_{n}$ and critical GJMS operator $P_{n}$ is “pure Ricci” , while any other
construct that shares the same conformal properties is “contaminated” by
explicit dependence on the Weyl tensor.

##  4 6D CFT: examples

There are few examples in 6D, worked out some years ago [24], that can now be
used to verify the correctness of the novel relation put forward above between
Casimir energy and trace anomaly coefficients. We first write down the trace
anomaly with the relevant coefficients, and in the table below we report the
values for the Casimir energies computed with the relation we found out and
the outcomes in all cases exactly match the known results in the literature
(see, e.g., [17]).

#### Conformal scalar:

The first instance is of course the conformally coupled scalar field, for
which we have

$(4\pi)^{3}\cdot 7!\,\langle T\rangle=\frac{5}{9}\cdot E_{6}-\frac{28}{3}\cdot
I_{1}+\frac{5}{3}\cdot I_{2}+2\cdot I_{3}+\nabla_{i}\cdot J^{i}~{},$ (29)

with the trivial total derivative term given by

$\displaystyle\nabla_{i}\cdot J^{i}$ $\displaystyle=$
$\displaystyle\frac{6}{5}\cdot C_{1}-\frac{2}{5}\cdot C_{2}+4\cdot
C_{3}+\frac{12}{5}\cdot C_{4}+\frac{17}{5}\cdot C_{6}+12\cdot C_{7}~{},$ (30)
$\displaystyle=$ $\displaystyle M_{5}-4\cdot M_{7}+7\cdot
M_{8}+\frac{1}{10}\cdot M_{9}+\frac{6}{5}\cdot M_{10}~{}.$

#### Massless Dirac squared:

The second instance of a conformally invariant free field corresponds to the
(square of the) massless Dirac operator, for which we have

$(4\pi)^{3}\cdot 7!\,\langle T\rangle=\frac{191}{9}\cdot
E_{6}-\frac{896}{3}\cdot I_{1}-32\cdot I_{2}+40\cdot I_{3}+\nabla_{i}\cdot
J^{i}~{},$ (31)

with the trivial total derivative term given this time by

$\displaystyle\nabla_{i}\cdot J^{i}$ $\displaystyle=$ $\displaystyle 24\cdot
C_{1}-\frac{148}{15}\cdot C_{2}+136\cdot C_{3}+48\cdot C_{4}-168\cdot
C_{5}+96\cdot C_{6}+352\cdot C_{7}~{},$ (32) $\displaystyle=$
$\displaystyle\frac{88}{3}\cdot M_{5}-136\cdot M_{7}+\frac{112}{3}\cdot
M_{8}-5\cdot M_{9}+24\cdot M_{10}~{}.$

#### Two-form:

The third instance of a conformally invariant free field corresponds to a two-
form after a covariant gauge-fixing, for which we have

$(4\pi)^{3}\cdot 7!\,\langle T\rangle=442\cdot E_{6}-\frac{8008}{3}\cdot
I_{1}-\frac{2378}{3}\cdot I_{2}+180\cdot I_{3}+\nabla_{i}\cdot J^{i}~{},$ (33)

and with the trivial total derivative term

$\displaystyle\nabla_{i}\cdot J^{i}$ $\displaystyle=$ $\displaystyle-60\cdot
C_{1}+\frac{2036}{15}\cdot C_{2}-1152\cdot C_{3}-120\cdot C_{4}-504\cdot
C_{5}-646\cdot C_{6}+856\cdot C_{7}~{},$ (34) $\displaystyle=$
$\displaystyle\frac{214}{3}\cdot M_{5}+312\cdot M_{7}-\frac{14}{3}\cdot
M_{8}+93\cdot M_{9}-60\cdot M_{10}~{}.$

In summary,

conformal field | $7!\cdot a$ | $7!\cdot g_{5}$ | $7!\cdot g_{6}$ | $7!\cdot g_{7}$ | $7!\cdot g_{8}$ | $7!\cdot g_{9}$ | $7!\cdot g_{10}$ | $E_{o}$
---|---|---|---|---|---|---|---|---
conformal scalar | $\frac{5}{9}$ | 1 | 0 | -4 | 7 | $\frac{1}{10}$ | $\frac{6}{5}$ | $-\frac{31}{60480}$
massless Dirac squared | $\frac{191}{9}$ | $\frac{88}{3}$ | 0 | -136 | $\frac{112}{3}$ | -5 | 24 | $-\frac{367}{24192}$
two-form | 442 | $\frac{214}{3}$ | 0 | 312 | $-\frac{14}{3}$ | 93 | -60 | $-\frac{191}{2016}$

As a verification of the above result for the conformal scalar in 6D, we can
evaluate directly in the Polyakov formula obtained by Branson [21] (see also
[30])

$\displaystyle-$ $\displaystyle\frac{3\cdot 7!\,(4\pi)^{3}}{2}$
$\displaystyle\log\frac{\det\hat{Y}}{\det
Y}=10\int_{M_{6}}dv_{g}\;\sigma\cdot\left\\{Q_{6}+\frac{1}{2}\,P_{6}\,\sigma\right\\}+13\left\\{\int_{\hat{M}_{6}}dv_{\hat{g}}\;\;|\hat{\nabla}\hat{J}|^{2}-\int_{M_{6}}dv_{g}\;|{\nabla}{J}|^{2}\right\\}$
(35)
$\displaystyle+34\left\\{\int_{\hat{M}_{6}}dv_{\hat{g}}\;\hat{J}^{3}-\int_{M_{6}}dv_{g}\;J^{3}\right\\}-32\left\\{\int_{\hat{M}_{6}}dv_{\hat{g}}\;\hat{J}\,|\hat{V}|^{2}-\int_{M_{6}}dv_{g}\;J\,|V|^{2}\right\\}~{},$

in terms of the Schouten tensor $V$ and scalar $J$. Likewise, for the
conformally related metrics we get

$\displaystyle-{3\cdot 7!\,(4\pi)^{3}}\,\beta\cdot E_{o}$ $\displaystyle=$
$\displaystyle
5\int_{\text{flatland}}dv_{g}\;|\nabla\,\nabla^{2}\,\sigma|^{2}+34\int_{S^{1}_{\beta}\times
S^{5}}dv_{\hat{g}}\;\hat{J}^{3}\,-\,32\int_{S^{1}_{\beta}\times
S^{5}}dv_{\hat{g}}\;\hat{J}\,|\hat{V}|^{2}$ (36) $\displaystyle=$
$\displaystyle\beta\,\pi^{3}\left\\{5\cdot 64\,+\,34\cdot 8\,-\,32\cdot
3\right\\}\,=\,496\,\beta\,\pi^{3}~{},$

and we again obtain $E_{o}=-\frac{31}{60480}$.

For the Dirac operator, we start with

$\displaystyle-\frac{9\cdot
7!\,(4\pi)^{3}}{2}\,\log\frac{\det\hat{\not{\nabla}}^{2}}{\det{\not{\nabla}}^{2}}$
$\displaystyle=$
$\displaystyle-1146\int_{M_{6}}dv_{g}\;\sigma\cdot\left\\{Q_{6}+\frac{1}{2}\,P_{6}\,\sigma\right\\}$
$\displaystyle-$ $\displaystyle
507\left\\{\int_{\hat{M}_{6}}dv_{\hat{g}}\;\;|\hat{\nabla}\hat{J}|^{2}-\int_{M_{6}}dv_{g}\;|{\nabla}{J}|^{2}\right\\}$
$\displaystyle-$ $\displaystyle
1578\left\\{\int_{\hat{M}_{6}}dv_{\hat{g}}\;\hat{J}^{3}-\int_{M_{6}}dv_{g}\;J^{3}\right\\}$
$\displaystyle+$ $\displaystyle
1752\left\\{\int_{\hat{M}_{6}}dv_{\hat{g}}\;\hat{J}\,|\hat{V}|^{2}-\int_{M_{6}}dv_{g}\;J\,|V|^{2}\right\\}~{},$

and thus for the conformally related metrics we obtain

$\displaystyle{9\cdot 7!\,(4\pi)^{3}}\,\beta\cdot E_{o}$ $\displaystyle=$
$\displaystyle-573\int_{\text{flatland}}dv_{g}\;|\nabla\,\nabla^{2}\,\sigma|^{2}-1578\int_{S^{1}_{\beta}\times
S^{5}}dv_{\hat{g}}\;\hat{J}^{3}\,+\,1752\int_{S^{1}_{\beta}\times
S^{5}}dv_{\hat{g}}\;\hat{J}\,|\hat{V}|^{2}$ (38) $\displaystyle=$
$\displaystyle\beta\,\pi^{3}\left\\{-573\cdot 64\,-\,1578\cdot 8\,+\,1752\cdot
3\right\\}\,=\,-44040\,\beta\,\pi^{3}~{},$

implying $E_{o}=-\frac{367}{24192}$.

##  5 Summary and outlook

In this work, the relation between the 6D CFT trace anomaly and the Casimir
energy due to compactifying the spatial section on a five-sphere has been
fully established. The resulting expression extends the 4D result of Cappelli
and Coste to 6D by taking account of the trivial total derivatives in the
trace anomaly. The scheme-dependence of the Casimir energy is now controlled
by the trivial total derivatives, or equivalently, by the finite counter-terms
that can be added to the effective action. We expect this will allow a better
understanding of the discrepancies in the vacuum energy in holographic
scenarios. The present relation would also be well suited to further explore
the connection, within $\zeta$-function regularization, between Casimir energy
and multiplicative anomaly [31] beyond conformal scalar fields 111111There is
a misprint in eqn. 8.3 of [31], the coefficient of $\gamma_{3}$ should be 1
and not 11. The total derivative $\nabla^{2}|V|^{2}$ must also be completed to
$\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)$ to make it a trivial total
derivative..
Let us conclude by mentioning that going beyond the conformally flat case
seems to be quite a challenge (see e.g. [32]). One of the obvious difficulties
consists in the choice of a fiducial metric, a suitable candidate could be the
homogeneously squashed (or Berger), instead of the round, sphere where many
explicit calculations are doable (see e.g. [33]).

We thank F. Bastianelli, L. Casarin, J.S. Dowker, E. Friedman, L. Peterson and
I.L. Shapiro for valuable conversations and comments. This work was partially
funded through FONDECYT-Chile 1220335. We have benefited from the Cadabra
software [34] to evaluate curvature invariants and the Mathematica package
xAct [35] to verify their conformal transformations.

##  Appendix A 6D basis of curvature invariants

Curvature invariants

$\displaystyle K_{1}=R^{3}\quad,\quad K_{2}=R\,Ric^{2}\quad,\quad
K_{3}=R\,Riem^{2}\quad,\quad K_{4}=Ric^{3}\quad,\quad K_{5}=-Riem\,Ric^{2}$
$\displaystyle K_{6}=Ric\,Riem\,Riem\quad,\quad K_{7}=Riem^{3}\quad,\quad
K_{8}=Riem^{\prime 3}\quad,\quad K_{9}=R\,\nabla^{2}\,R$ $\displaystyle
K_{10}=Ric\,\nabla^{2}\,Ric\quad,\quad
K_{11}=Riem\,\nabla^{2}\,Riem\quad,\quad K_{12}=Ric\,\nabla\nabla R\quad,\quad
K_{13}=|\nabla Ric|^{2}$ $\displaystyle K_{14}=\nabla Ric\nabla Ric\quad,\quad
K_{15}=|\nabla Riem|^{2}\quad,\quad K_{16}=\nabla^{2}R^{2}\quad,\quad
K_{17}=\nabla^{4}R~{}.$ (39)

Trivial anomalies:

$\displaystyle M_{5}$ $\displaystyle=$ $\displaystyle
6K_{6}-3K_{7}+12K_{8}+K_{10}-7K_{11}-11K_{13}+12K_{14}-4K_{15}~{},$ (40)
$\displaystyle M_{6}$ $\displaystyle=$
$\displaystyle-\frac{1}{5}K_{9}+K_{10}+\frac{2}{5}K_{12}+K_{13}~{},$
$\displaystyle M_{7}$ $\displaystyle=$ $\displaystyle
K_{4}+K_{5}-\frac{3}{20}K_{9}+\frac{4}{5}K_{12}+K_{14}~{},$ $\displaystyle
M_{8}$ $\displaystyle=$
$\displaystyle-\frac{1}{5}K_{9}+K_{11}+\frac{2}{5}K_{12}+K_{15}~{},$
$\displaystyle M_{9}$ $\displaystyle=$ $\displaystyle K_{16}$ $\displaystyle
M_{10}$ $\displaystyle=$ $\displaystyle K_{17}.$

Conformal primitives 121212These relations were reported in appendix C of
[24]. We are indebted to F. Bastianelli for pointing this out to us.:

$\displaystyle\mathcal{F}_{5}=\frac{1}{30}K_{1}-\frac{1}{4}K_{2}+K_{6}\quad,\quad\mathcal{F}_{6}=\frac{1}{100}K_{1}-\frac{1}{20}K_{2}\quad,$
$\displaystyle\mathcal{F}_{7}=\frac{37}{6000}K_{1}-\frac{7}{150}K_{2}+\frac{1}{75}K_{3}-\frac{1}{10}K_{5}-\frac{1}{15}K_{6}\quad,\quad\mathcal{F}_{8}=\frac{1}{150}K_{1}-\frac{1}{20}K_{3}\quad,$
$\displaystyle\mathcal{F}_{9}=-\frac{1}{30}K_{1}\quad,\quad\mathcal{F}_{10}=\frac{1}{300}K_{1}-\frac{1}{20}K_{9}\quad.$
(41)

Curvature invariants at $S^{1}_{\beta}\times S^{5}$:

$\displaystyle K_{1}=R^{3}=8000\quad,\quad K_{2}=R\,Ric^{2}=1600\quad,\quad
K_{3}=R\,Riem^{2}=800\quad,$ $\displaystyle
K_{5}=-Riem\,Ric^{2}=-320\quad,\quad K_{6}=Ric\,Riem^{2}=160\quad,\quad
K_{9}=R\,\nabla^{2}\,R=0\quad.$ (42)

##  Appendix B 6D Branson’s basis of conformal primitives

On conformally flat manifolds the general basis of six trivial total
derivatives is reduced to only three independent terms, stemming from only
three conformal primitives. One convenient choice of conformal primitives
consists of $J^{3},|\nabla J|^{2}$ and $J|V|^{2}$. To compute the trivial
total derivatives obtained by the conformal variation of these curvature
invariants under $g\rightarrow g_{\omega}=e^{2\omega}g$, it is enough to keep
only the term linear in the conformal factor $\omega$ and discard boundary
terms since the manifold is closed.
Let us start with $J^{3}$

$\int_{\hat{M}_{6}}dv_{\omega}J_{\omega}^{3}=\int_{M_{6}}dv(J-\nabla^{2}\omega)^{3}=\int_{M_{6}}dv(J^{3}-3J^{2}\nabla^{2}\omega)\;\rightarrow\;\int_{M_{6}}dv\,\omega(-3\nabla^{2}J^{2})~{},$
(43)

so that $J^{3}$ turns out to be a conformal primitive for $-3\nabla^{2}J^{2}$.
In the case of $|\nabla J|^{2}$, as conformal primitive it is equivalent to
$-J\nabla^{2}J$ since they add up to a boundary term
$\frac{1}{2}\nabla^{2}J^{2}$. A shortcut to compute the conformal variation is
provided by considering the Yamabe operator $Y=-\nabla^{2}+2J$ instead of the
plain Laplacian, and exploit the conformal covariance of $Y$ as follows

$\displaystyle\int_{\hat{M}_{6}}dv_{\omega}J_{\omega}Y_{\omega}J_{\omega}=\int_{M_{6}}dv(J-\nabla^{2}\omega)$
$\displaystyle Y$
$\displaystyle(J-\nabla^{2}\omega)=\int_{M_{6}}dv\left\\{JYJ-(\nabla^{2}\omega)YJ-
JY\nabla^{2}\omega)\right\\}$ $\displaystyle\rightarrow$
$\displaystyle\;\int_{M_{6}}dv\,\omega(-2\nabla^{2}YJ)~{}.$ (44)

The integration by parts performed above is equivalent to the self-adjointness
of both the Laplacian and the Yamabe operator. We end up with the conclusion
that a conformal primitive for $-2\nabla^{2}YJ$ is given by
$JYJ=J(-\nabla^{2}+2J)J$ or, equivalently, $|\nabla J|^{2}+2J^{3}$. Combining
this with the previous result for $J^{3}$, we obtain that $|\nabla J|^{2}$ is
a conformal primitive for $2\nabla^{2}\nabla^{2}J+2\nabla^{2}J^{2}$.
A shortcut to compute the conformal variation of the remaining curvature
invariant $J|V|^{2}$ involves the Paneitz operator P acting on the Schouten
scalar

$\displaystyle\int_{\hat{M}_{6}}dv_{\omega}Q_{4,\omega}J_{\omega}=\int_{\hat{M}_{6}}dv_{\omega}P_{\omega}J_{\omega}$
$\displaystyle=$
$\displaystyle\int_{M_{6}}dve^{\omega}Pe^{-\omega}(J-\nabla^{2}\omega)$
$\displaystyle=\int_{M_{6}}dv\left\\{PJ-P\nabla^{2}\omega+e^{\omega}P^{o}(e^{-\omega}J)\right\\}$
$\displaystyle=$
$\displaystyle\int_{M_{6}}dv\left\\{Q_{4}J-Q_{4}\nabla^{2}\omega+JP^{o}\omega\right\\}$
$\displaystyle\rightarrow$
$\displaystyle\;\int_{M_{6}}dv\,\omega(-\nabla^{2}Q_{4}+P^{o}J)~{}.$ (45)

We have kept only terms at most linear in $\omega$ and made use of the
following properties of the Q-curvature $Q_{4}$ and Paneitz operator $P$ in 6D

$P=P^{o}+Q_{4}\;,\;Q_{4}=3J^{2}-2|V|^{2}-\nabla^{2}J\;,\;P_{\omega}=e^{-5\omega}Pe^{\omega},$
(46)

where $P^{o}$ denotes the total derivative part of the Paneitz operator

$P^{o}=\nabla^{2}\nabla^{2}+4\nabla\cdot\left(V\cdot\;-\;J\right)\nabla~{}.$
(47)

We have therefore that $Q_{4}J$ is a conformal primitive of
$-\nabla^{2}Q_{4}+P^{o}J$ which equals
$2\nabla^{2}\nabla^{2}J-5\nabla^{2}J^{2}+2\nabla^{2}|V|^{2}+4\nabla\cdot(V\cdot\nabla
J)$. With all the instances gathered so far, we can finally disentangle the
$-J|V|^{2}$ term in $Q_{4}J$, and obtain that it is a conformal primitive for
$\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)+\nabla^{2}J$ . We summarize
the above results, which have also been independently verified with the xAct
package for Mathematica, in the table below.

trivial total derivative |  conformal primitive
---|---
$\nabla^{2}\nabla^{2}J$ | $\frac{1}{2}|\nabla J|^{2}+\frac{1}{3}J^{3}$
$\nabla^{2}J^{2}$ | $-\frac{1}{3}J^{3}$
$\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)$ | $-J|V|^{2}+\frac{1}{3}J^{3}$

Endowed with this correspondence, we can now determine the contribution to the
partition function that originates in the trivial total derivatives of the
trace anomaly on conformally flat backgrounds

$\displaystyle(4\pi)^{3}\cdot\langle T\rangle$ $\displaystyle=$
$\displaystyle-a_{6}\cdot
E_{6}+\gamma_{1}\nabla^{2}\nabla^{2}J+\gamma_{2}\nabla^{2}J^{2}+\gamma_{3}\\{\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla
J)\\}$ (48) $\displaystyle=$ $\displaystyle-6\,a_{6}\cdot
Q_{6}+\tilde{\gamma}_{1}\nabla^{2}\nabla^{2}J+\tilde{\gamma}_{2}\nabla^{2}J^{2}+\tilde{\gamma}_{3}\\{\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla
J)\\}~{}.$

The contribution to the partition function from the trivial total derivatives
reduces to the evaluation on the cylinder of their corresponding conformal
primitives. Collecting similar terms, evaluating the conformal primitives
$J^{3}=8$, $|\nabla J|^{2}=0$ and $J|V|^{2}=3$, we arrive at volume times a
particular combination of coefficients

$\frac{\beta\,\pi^{3}}{64\pi^{3}}\left\\{\frac{8\gamma_{1}}{3}-\frac{8\gamma_{2}}{3}+\frac{8\gamma_{3}}{3}-3\gamma_{3}\right\\}=\frac{\beta}{24}\left\\{\gamma_{1}-\gamma_{2}-\frac{1}{8}\gamma_{3}\right\\}~{},$
(49)

so that we obtain for the Casimir energy in the Euler basis

$E_{o}=-\frac{15}{8}a_{6}-\frac{1}{24}\left(\gamma_{1}-\gamma_{2}-\frac{1}{8}\gamma_{3}\right)~{},$
(50)

or, alternatively, in the Q-curvature basis

$E_{o}=-3a_{6}-\frac{1}{24}\left(\tilde{\gamma}_{1}-\tilde{\gamma}_{2}-\frac{1}{8}\tilde{\gamma}_{3}\right)~{}.$
(51)

The degeneracy of the basis for total derivatives at conformally flat
manifolds is summarized in the table below.

general basis of t.t.d. |  reduced basis of t.t.d.
---|---
$M_{5}$ | $-24\\{\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)\\}-7\nabla^{2}J^{2}$
$M_{6}$ | $8\\{\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)\\}-\nabla^{2}J^{2}$
$M_{7}$ | $8\\{\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)\\}-\frac{7}{2}\nabla^{2}J^{2}$
$M_{8}$ | $8\\{\nabla^{2}|V|^{2}+2\nabla\cdot(V\cdot\nabla J)\\}-6\nabla^{2}J^{2}$
$M_{9}$ | $100\nabla^{2}J^{2}$
$M_{10}$ | $10\nabla^{2}\nabla^{2}J$

This dictionary allows a direct verification of the vanishing on conformally
flat manifolds of the freedom in Anselmi’s pondered Euler density (eqn.24) and
the two freedoms found out by Hamada (eqns.27 and 28). It also allows to
verify the relation for the Casimir energy in terms of the coefficients of the
reduced basis, collecting their coefficients

$\displaystyle E_{o}$ $\displaystyle=$
$\displaystyle-\frac{15}{8}a_{6}-\frac{1}{24}\left([10g_{10}]-[-7g_{5}-g_{6}-\frac{7}{2}g_{7}-6g_{8}+100g_{9}]-\frac{1}{8}[-24g_{5}+8g_{6}+8g_{7}+8g_{8}]\right)$
(52) $\displaystyle=$
$\displaystyle-\frac{15}{8}a_{6}-\frac{5}{12}\left(g_{5}+\frac{1}{4}g_{7}+\frac{1}{2}g_{8}-10g_{9}+g_{10}\right)~{}.$

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