# Real-space Green’s function approach for x-ray spectra at finite temperature

Tun S. Tan<EMAIL_ADDRESS>J. J. Kas  J. J. Rehr  Dept. of Physics, Univ. of
Washington, Seattle, WA 98195-1560

###### Abstract

There has been considerable interest in properties of condensed matter at
finite temperature, including non-equilibrium behavior and extreme conditions
up to the warm dense matter regime. Such behavior is encountered, e.g., in
experimental time-resolved x-ray absorption spectroscopy (XAS) in the presence
of intense laser fields. In an effort to simulate such behavior, we present an
approach for calculations of finite-temperature x-ray absorption spectra in
arbitrary materials, using a generalization of the real-space Green’s function
formalism. The method is incorporated as an option in the core-level x-ray
spectroscopy code FEFF10. To illustrate the approach, we present calculations
for several materials together with comparisons to experiment and with other
methods.

###### pacs:

##  I Introduction

X-ray absorption spectroscopy (XAS) has become an important tool for studies
of materials in fields ranging from materials science and chemistry to
geophysics and astrophysics [1, 2, 3, 4, 5, 6]. XAS is used extensively to
probe both local electronic and structural properties simultaneously at
synchrotron facilities worldwide. With the development of x-ray free electron
lasers (XFELs), XAS experiments have been extended to treat ultra-short femto-
to picosecond time-scales and non-equilibrium conditions, with temperatures
$T$ up to many thousands of K. These capabilities are important, e.g., for
studies of matter in extreme conditions as well as non-equilibrium and dynamic
response due to electron-phonon energy transfer, spin-relaxation and
reactions, as well as in shocked matter. The extreme conditions include the
warm dense matter (WDM) regime, where $T\sim T_{F}$ (the Fermi temperature)
and the density is within an order of magnitude or so of normal conditions,

Modern theories of optical and x-ray spectra start from the many-body Fermi’s
golden rule and are usually cast at zero temperature. In order to make the
calculations computationally tractable, a single-particle or quasi-particle
approximation for the photoelelectron is often used [7, 8]. The quasi-particle
approximation is the basis of the real-space Green’s function (RSGF) approach
used in the FEFF codes [9], which are applicable to systems throughout the
periodic table. Our main aim in this work is to extend the RSGF approach for
XAS to finite temperature (FT). Secondly we aim to explore FT and non-
equilibrium behavior in a few systems that can be measured experimentally. Our
FT generalization of the RSGF approach requires a number of extensions, since
many electronic ingredients, e.g., the chemical potential, exchange-
correlation potential, quasi-particle self-energy, and mean-free path are
temperature dependent. Vibrational effects are also temperature dependent and
strongly damp the fine-structure in XAS at high $T$. At low temperatures
compared to the Fermi temperature $T<<T_{F}$ (which is typically a few $\times
10^{4}$ K), the exchange-correlation potential and self-energy are weakly
temperature dependent and a zero-temperature approximation is often adequate
for electronic structure. Vibrational effects become substantial when $T$ is
of order the Debye temperature $T_{D}$ (which is typically a few $\times
10^{2}$ K). However, in the WDM regime where $T\sim T_{F}$, an explicit
account of temperature dependence becomes necessary, since the exchange-
correlation potential varies from exchange- to correlation-dominated behavior
with increasing temperature in that regime [10]. At finite-temperature an
efficient approximation for the XAS cross-section can be obtained using the
RSGF formalism, which is the real-space analog of the KKR band structure
approach [11]. As at $T=0$, the approach is based on the one-particle electron
Green’s function, which is calculated either by matrix inversion or by a well
converged multiple-scattering path expansion. Our current implementation
builds upon the theory and algorithms used in the FEFF9 code [9], and has been
incorporated in a new version FEFF10 [12]. This theory is illustrated here
with a number of examples. While the theory is generally applicable to
periodic or aperiodic systems alike throughout the periodic table, we focus on
applications at normal densities, where the theory can be tested against
recent experiment.

The remainder of the paper is organized as follows: in Sec. II. we review the
quasiparticle XAS theory and present the RSGF formalism. Sec. III. treats
finite temperature effects and Sec. IV. presents a number of examples. Finally
Sec. V. contains a brief summary and conclusions.

##  II  Finite temperature XAS

In this section, we briefly summarize the generalization of quasiparticle
theory of XAS to finite temperature. Theories of the XAS cross section
typically start from the many-body Fermi golden rule

$\displaystyle\sigma(\omega)=4\pi^{2}\frac{\omega}{c}\sum_{F}|\langle
I|\hat{D}|F\rangle|^{2}\delta(\omega+E_{I}-E_{F}),$ (1)

where $|I\rangle$ is the many-body ground state of the system with total
energy $E_{I}$, $|F\rangle$ ranges over the many-body final states with energy
$E_{F}$, and $\hat{D}$ is the many-body transition operator due to the x-ray
field, which here is taken to be a dipole-interaction
${D}=\Sigma_{i,f}d_{ij}\hat{c}^{\dagger}_{f}\hat{c}_{i}$ where $d_{ij}$ are
the dipole matrix elements, and $c_{i}$ and $c_{i}^{\dagger}$ are electron
annihilation and creation operators respectively. Here and throughout this
paper we use atomic units $e=\hbar=m=1$, and units of temperature where
$k_{B}=1$, unless noted otherwise. In order to reduce the calculation to a
quasi-particle framework, we make a sudden approximation as in the $\Delta$SCF
approximation, with the final-state rule [13]. Since a core electron leaves
behind a hole after being excited into the photo-electron state the final
photo-electron state $|\phi_{f}\rangle$ must take into account the interaction
with the core hole potential, while a given core-level $|\phi_{i}\rangle$ is
calculated with the initial state configuration. Next the many-particle
initial and final states are factored such that
$|I\rangle=|\phi_{i}\rangle|\Phi^{N-1}_{0}\rangle$ and
$|F\rangle=|\phi_{f}\rangle|\tilde{\Phi}^{N-1}_{n}\rangle$, where
$|\tilde{\Phi}^{N-1}_{n}\rangle$ is the $n^{th}$ excited state of the $N-1$
electron system with a core-hole in level $i$, and $|\phi_{f}\rangle$ is the
photo-electron state calculated in the presence of the core-hole. If one
ignores the energy dependence of excitations of the $N-1$ electron system, Eq.
(1) yields an effective single- or quasi-particle cross section
$\sigma_{1}(\omega)$ contribution from a given core level $i$ given by

$\displaystyle\sigma_{1}(\omega)=4\pi^{2}\frac{\omega}{c}\sum_{f}|\langle
i|\hat{d}|f\rangle|^{2}\delta_{\Gamma}(\omega+\epsilon_{i}-\epsilon_{f}),$ (2)

where $\epsilon_{i}$ and $\epsilon_{f}$ are the eigenenergies of the
quasiparticle initial $|i\rangle$ (deep-core) and final $|f\rangle$ (photo-
excited) states, and $\delta_{\Gamma}(\omega)$ denotes a Lorentzian of width
$\Gamma$, where $\Gamma$ includes the lifetime and final-state broadening.
More generally this approximation can be corrected by a convolution of the
quasi-particle XAS with the core spectral function, defined as
$A_{c}(\omega)=\Sigma_{n}S_{n}^{2}\delta(\omega-\omega_{n})$ where
$S_{n}^{2}=|\langle\Phi^{N-1}_{0}|\tilde{\Phi}^{N-1}_{n}\rangle|^{2}$ is the
many-body amplitude overlap integral between initial and final states. This
convolution modifies the cross-section by an additional, energy dependent
broadening factor [14], of unit weight which damps the fine structure by an
constant factor, which historically has been denoted by $S_{0}^{2}$ [15].

###  II.1 Real-space Green’s function

Formally the retarded single electron Green’s function $G^{R}(\epsilon)$ in a
basis of local site, angular momentum states[15] $|L,j\rangle=j_{L}({\bf
r})Y_{L}(\hat{r})$ is given by the spectral sum

$\displaystyle G_{L^{\prime}j^{\prime},Lj}(\epsilon)$ $\displaystyle=$
$\displaystyle\langle L^{\prime}j^{\prime}|G^{R}(\epsilon)|L,j\rangle$ (3)
$\displaystyle=$ $\displaystyle\sum_{f}\frac{\langle
L^{\prime}j^{\prime}|f\rangle\langle
f|Lj\rangle}{\epsilon-\epsilon_{f}+i\delta},$

where $\varepsilon_{f}$ is the eigenenergy of the final state SCF quasi-
particle Hamiltonian $H^{\prime}=p^{2}/2+v_{f}+\Sigma(\epsilon;T)$, $v_{f}$ is
the effective one electron Coulomb potential in the presence of a screened
core hole, and $\Sigma(\epsilon)$ is the dynamically screened FT electron
self-energy. A significant advantage of the RSGF formalism is that it
implicitly builds in the summation over final states $f$, which is a
computational bottleneck in wave-function approaches at high energies [9]. In
FEFF the RSGF approach implicitly includes Dirac-relativistic effects, and the
angular momentum index $L$ denotes the relativistic angular quantum numbers
$L=(\kappa,m)$[15]. In the following temperature dependence will be assumed
implicitly, unless otherwise specified. By inserting the spectral
representation of $G^{R}(\epsilon)$ into Eq. (2) together with Fermi-Dirac
occupation numbers, the final states are implicitly summed in the calculation
of $\sigma_{qp}(\omega)$ using the solution to the Dyson equation [16], as
summarized below. Thus

$\sigma_{qp}(\omega)=L_{\Gamma}(\omega)*\sigma^{0}_{1}(\omega),$ (4)

where $L_{\Gamma}(\omega)$ is a Lorentzian function that accounts for the
core-hole lifetime broadening and $\sigma_{1}^{0}$ is the single- or
quasiparticle cross-section with no core-hole broadening, given by

$\displaystyle\sigma_{qp}^{0}(\omega)$ $\displaystyle=$
$\displaystyle-4\pi\frac{\omega}{c}\,\textnormal{Im}\sum_{ill^{\prime}}\langle
i|\hat{d}\ G^{R}_{l0,l^{\prime}0}(\omega+\epsilon_{i})\
\hat{d}^{\dagger}|i\rangle$ (5) $\displaystyle\times
f_{T}(\epsilon_{i})\big{[}1-f_{T}(\omega+\epsilon_{i})\big{]},$

where site 0 is taken to be the absorbing atom,
$f_{T}=1/[\exp(\epsilon-\mu(T))+1]$ is the Fermi-Dirac distribution, and
$\mu(T)$ is the chemical potential of the system. Much of the electronic
temperature dependence is due to that in the occupation numbers $f_{T}$.

Multiple-scattering of the photoelectron by its environment is naturally built
into the RSGF formalism[16]. The intra-atomic contributions from the absorbing
atom $G_{c}$ can be separated from the scattering atoms $G_{sc}$ into
$G^{R}=G^{R}_{c}+G^{R}_{sc}$. (For simplicity, here and below we drop the
subscripts on matrix elements of $G$.) The scattering contribution
$G^{R}_{sc}$ is determined by the free Green’s function $G^{R}_{0}$, the total
scattering matrix $T$, and partial-wave phase shifts $\delta$ for the
photoelectron state of a given angular momentum (suppressed) using the matrix
inverse solution to the Dyson equation [13].

$G^{R}_{sc}=e^{i\delta}\big{(}1-G^{R}_{0}T\big{)}^{-1}G^{R}_{0}e^{i\delta^{\prime}}.$
(6)

For EXAFS energies (typically above about 50 eV beyond an absorption edge),
the matrix inverse can be expanded in a rapidly converging series of
scattering paths typically with less than 4 legs. However, for XANES, full
multiple scattering (FMS) is usually needed, and can be carried out by a fast
matrix inversion algorithm since the basis of relevant angular-momentum, site
scattering states is small due to the short mean free path and $l_{max}<4$.

##  III Finite-temperature effects

Several effects come into play in the theory of XAS at finite temperatures.
First, there are effects of electronic temperature in the initial many-body
electronic configuration of the system due to the temperature dependence of
the electron density and chemical potential. Second, the Fermi function cutoff
that defines the x-ray “edge” of the cross-section as in Eq. (5) broadens with
increasing $T$. Third there are FT effects on the exchange and correlation,
both through the exchange-correlation potential $v_{xc}$ and the quasi-
particle self-energy $\Sigma(\epsilon)$. Finally, there are the FT effects of
lattice vibrations which give rise to Debye-Waller factors that strongly damp
the XAS fine-structure. These effects are summarized below:

1) FT SCF: As for the ground state, in order to construct the (scattering)
potential $v$, we implement a generalization of the self-consistent field
(SCF) method to calculate the Coulomb potentials, electron densities and a
temperature dependent chemical potential. In the RSGF approach in FEFF, the
SCF procedure starts with the overlapped atomic electron densities obtained by
solving the relativistic Dirac-Fock equations for each atom in the system.
From this initial guess of the electron density, a Green’s function is
calculated, which provides a new density and chemical potential. This
procedure is then repeated until self-consistency ($\rho_{i}=\rho_{i-1}$) is
reached to high accuracy, typically in about 20 iterations or so. Assuming a
frozen core, the valence electron contribution to density of electrons is
given in terms of an integral over energy of the imaginary part of the Green’s
function,

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi}\int_{E_{B}}^{\infty}dE\ \textnormal{Im}\
G(r,r,E)f_{T}(E),$ (7)

where $E_{B}$ is the core-valence separation energy (typically set to -40 eV
in FEFF), and the factor of 2 accounts for spin-degeneracy.

Figure 1: Energy contour for electron density calculation at finite
temperature: $\omega_{j}=\pi(2j-1)kT$ denotes the imaginary part of the
Matsubara poles of the fermi function $f(\omega)$. The integration starts from
some core-valence separation level $E_{B}$ above the core states but below the
valence states. Contour A (blue) represents the original integration path just
above the real axis, while contour C (red) represents the integration path in
the complex plane.

For computational efficiency at both zero and finite temperatures, the above
integral is carried out in the complex plane where the Green’s function
becomes smooth. However, there is an added complication due to the possible
presence of the Matsubara poles in the Fermi function. In order to include
them, we integrate along a contour C (shown in Fig. 1), where the first leg
traverses from $E_{B}$ to $E_{B}+2\pi inkT$, then to $\infty+2\pi inkT$, where
$2\pi inkT$ is halfway between the Matsubara poles. For a given chemical
potential, the valence density is then given by

$\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle-\frac{2}{\pi}{\rm
Im}\,\Bigg{[}\int_{C}dE\ \ G(r,r,E)f_{T}(E),$ (8) $\displaystyle-2\pi
ikT\sum_{j=1}^{n}G(r,r,z_{j})\Bigg{]},$

where $z_{j}=\mu\pm i\pi(2j-1)kT$ are the Matsubara poles, and $n$ is the
number of poles between the contour C and the real axis. Eq. (8) also has
implicit dependence on the chemical potential through the Fermi function. In
turn the chemical potential is found by enforcing charge neutrality,

$N_{el}=\int dr\,\rho(r;\mu(T)),$ (9)

where $N_{el}$ is the number of valence electrons in the system, again
treating the core-electrons as frozen.

Eqs. (4) and (5) for the cross section contain similar integrals of the
Green’s function multiplied by the Fermi function. These integrals are dealt
with in a similar manner, although the contour is slightly different, in
principal extending from $-\infty+i2\pi nk_{B}T$ to $+\infty+i2\pi nk_{B}T$.
There is also an additional pole arising from the Lorentzian broadening
function $L_{\Gamma}(\omega)$.

2) FT exchange-correlation potentials: Additional temperature dependence comes
from that in the exchange-correlation potentials (which affect the initial
state densities and potentials). An approach for calculating these exchange-
correlation potentials from first principles with a FT cumulant Green’s
function approach has been discussed by Kas et al.[10] which shows that the
potentials cross-over from exchange- to correlation-dominated with increasing
$T$. Here, the behavior is treated by an efficient parameterized extension of
DFT fit to Quantum-Monte Carlo calculations (QMC) by Karasiev et al. [17],
which is in good agreement with the results of Kas et al. [10].

3) FT Self-energy effects: The temperature dependence of the dynamical
quasiparticle self-energy becomes important in calculations of the quasi-
particle photoelectron states and hence the transition matrix elements at
energies above an edge. Since full calculations of the FT self-energy[18] are
computationally demanding, we have used a much more efficient FT COHSEX
approximation to the GW self-energy for this purpose [19].

4) Lattice vibrations: As at $T=0$ Lattice vibrations strongly damp electron
scattering at high energy. Approximate calculations at temperatures up to
several times the Debye-temperature can be done by including Debye-Waller
factors $\exp(-2\sigma^{2}k^{2})$ in the Green’s function propagator or
multiple scattering paths. This can be done using the correlated Debye model
and mean square radial displacements $\sigma^{2}(T)\propto T/T_{D}$, where
$T_{D}$ is the Debye temperature. However, this approach becomes inapplicable
in the near edge regime (where symmetry breaking occurs) and at high
temperatures $T>>T_{D}$, where the quasi-harmonic approximation breaks down.
For those cases a configurational average is called for. At temperatures well
above $T_{D}$, one can use finite-$T$ molecular dynamics (MD) to obtain a
temperature dependent configurational average of the spectrum. Closer to zero
temperature, zero-point motion can also be included via Quantum-Monte-Carlo
sampling.

##  IV Calculations

###  IV.1 Simple Metal: Al

Aluminum (fcc Al) is a prototypical nearly-free-electron system for testing
electronic structure calculations. The electronic density of states (DOS) for
Al in the conduction band is similar to that for a free electron model, with a
square root like dispersion at the bottom of the band. We show the chemical
potential shift vs electron temperature $T_{e}$ for aluminum up to $T_{e}$ =
10 eV (1 eV = 11604 K) in Fig. 2.

Figure 2: Chemical potential shift for aluminum, $\mu(T)-\mu(0)$ versus
electron temperature, $T_{e}$ using the ground state von Barth-Hedin [20]
(blue) and finite $T$ KSDT [17] (red) exchange-correlation potentials. For
comparison the $T=0$ K DFT calculation [21] (orange) is also shown up to $T=$
4 eV. The result with the Sommerfeld expansion (green) is shown for KSDT
exchange-correlation potential. The inset shows the chemical potential shift
up to $T=$ 1 eV.

For $T=0$ we used the von Barth Hedin ($v_{xc}^{VBH}$) exchange-correlation
potential [20] and and for $T>0$ the temperature dependent approximation of
KSDT ($v_{xc}^{KSDT}$) [17]. Our $v_{xc}^{VBH}$ chemical potential agrees well
with the calculations of Lin et. al [21] up to $T=$ 4 eV, and is reasonably
accurate up to about 10 eV, consistent with the $T^{2}$ behavior of the
Sommerfeld expansion. Remarkably the Sommerfeld-expansion remains a good
approximation even at very high $T$ of order several eV.

Figure 3: K-edge XAS for aluminum vs electron temperature including only
temperature dependent KSDT exchange correlation potentials (see text). Note
that the edge broadens considerably with increasing temperature. Figure 4:
Calculated K-edge XAS for equilibrium Al ($T_{e}=T_{L}$) at solid density vs
experimental XAS at $T=$ 0.27 and 1.76 eV. From bottom to top, QMD-DFT are
simulated at $T=$ 0.43, 0.86 and 0.07 eV while FEFF simulation are done at
$T=$ 0.27, 1.4 and 3.0 eV. Note the substantial broadening of the edge at
increased $T$, roughly consistent with experiment, and that lattice vibration
effects completely damp the fine-structure. The fluctuations in the
experimental XAS reflect experimental noise.

Next, we investigate the temperature dependence of the K-edge XAS of Al at
normal density up to WDM temperatures. We show the calculated K-edge XAS for
electron temperatures $T_{e}$ = 0.025, 0.5, and 3.0 eV in Fig. 3. The behavior
at the edge is consistent with broadening of the Fermi-Dirac distribution due
to the nearly free electron density of states of Al. The “pre-edge” behavior
is due to the increasing contribution from previously occupied states below
the $T=0$ Fermi level.

In order to account for thermal vibrations, we use the correlated Debye model
($T_{D}$ = 430 K. For comparison we show DFT based XAS calculated using
quantum molecular dynamics (QMC) averaged over several QMD runs [22].
Investigations of the Al K-edge under isochoric heating ($T_{e}=T_{L}$) at
normal density were also done,[22] with measurements at temperatures
$T_{e}=T_{L}=$ 0.025, 0.09, 0.27, 1.40 and 1.76 eV, as in Fig. 4. Our
calculation with the correlated Debye model agrees with fairly well with these
results. In principle, the correlated Debye model breaks down at high
temperatures ($T>0.27$ eV) when anharmonicity becomes large, but this is not a
serious problem as the fine-structure in the XAS is largely suppressed at
these high temperatures.

###  IV.2 Warm Dense Cu

Being a noble transition metal, Cu has substantially different excited-state
properties compared to simple nearly free electron metals like aluminum, due
to the highly localized $d$-band just below the Fermi level. Hence, at
elevated temperatures, the XAS also differs. Several works [4, 23, 24, 25]
studied these changes in the $L_{2,3}$-edge of Cu XAS in WDM using a more
computationally expensive AIMD simulation. Here we investigate the XAS of Cu
using our SCF RSGF approach and the correlated Debye model.

Figure 5: $\textnormal{L}_{3}$-edge XAS of Cu at normal density for (Top)
electron temperature $T_{e}=$ 300 K up to 9000 K at lattice temperature
$T_{L}=$ 300 K; and (Bottom) for lattice temperature $T_{L}=$ 300 K up to 5000
K at electron temperature $T_{e}=$ 300 K. The DFT calculation [26] is shown
for $T_{e}=$ 300 K (light blue) and 5800 K (green). The lower panels show the
change in XAS with respect to normal conditions $T_{e}=T_{L}=$ 300 K.

Fig. 5 shows the simulated XANES spectra at normal density for various
temperatures. Our calculation is broadened with a Gaussian to match that in
the the DFT-based calculation [26]. As expected, the pre-edge structure in the
XAS can be attributed to the increasing contribution from the d-states as the
Fermi-Dirac distribution broadens with increasing $T$ [4, 23, 24, 25].
Conversely, states just above the Fermi level have a reduced contribution
leading to a smaller peak just above the edge. The changes in XAS due to
electronic temperature are mainly localized in the near edge, whereas that due
to the lattice temperature affects the region above the edge, especially the
fine structure.

Figure 6: (Top) Edge shift in the $\textnormal{L}_{3}$-edge of Cu as a
function of electron temperature (red); the sudden shift is due to the onset
of the $d$-band edge. (Bottom) Change in pre-edge area as a function of
electron temperature. The DFT calculation (green dashes) [26] was computed
using ABINIT at lattice temperature $T_{L}$ = 300 K. Note that in constrast to
the edge shift, the pre-edge area varies smoothly with temperature for Cu.

Figure 7: Dynamics of electron temperature (blue) and lattice temperature
(red) based on two-temperature model calculation. The absorption power is
calculated using Beer-Lambert law for a Gaussian profile 400 nm laser source
for a 70 nm thick copper.

.

The edge-shift and the pre-edge area are both candidates for assessing the
internal temperature of a system. In order to extract the edge shift and pre-
edge area, the edge position is defined to be the first local maximum of the
first derivative in the reference XANES spectrum. The electron temperature
dependence of the edge shift and pre-edge area are shown in Fig. 6. Note that
the pre-edge area is monotonically increasing with the electron temperature
and it becomes linearly correlated to $T_{e}$ above 6000 K, consistent with
previous results.[26] In contrast the edge-shift is highly non-linear,
exhibiting an abrupt shift at about 7000 K. Therefore, for Cu and probably for
other transition metals as well, the pre-edge area is a better proxy for an
electron temperature thermometer, compared to the edge shift.

Next, we compare our finite-temperature calculation to time-resolved, WDM
experiments [4]. The system being explored is a 70 nm copper foil heated
optically by a 400 nm laser at a fluence of 0.33 J/$\textnormal{cm}^{2}$. As a
consequence the electrons are optically excited leading to an initial huge
pre-edge peak below the $L_{3}$-edge. In order to model the temperature
evolution, we used a two-temperature (2T) model.[4] with the same parameters
for the electron heat capacity and electron-phonon coupling factor as in
Zhibin et al. [21]. The temperature evolution is shown in Fig. 7. Note that
the lattice temperature raises quickly above the melting temperature $\sim$
1358 K under 1 ps due to the strong electron-phonon coupling in copper. Our
simulations use XAS from 20 atomic configurations taken from QMD calculations
using the VASP code [27, 28] with generalized gradient approximation (GGA)
exchange-correlation potentials [29]. We also used the PAW potentials with an
energy-cutoff of 590 eV. The system with a 2$\times$2$\times$2 supercell
constructed from a conventional unit cell of 8 atoms is propagated with a time
step of 1 fs to reach equilibration, and the sampling of configurations is
performed by randomly sampling from a 2 ps long trajectory with a time step of
1 fs. We compare our simulation at temperatures $T_{e}$ = 300 K, 10200 K and
6000 K for the time-delays $t<$ 0 ps, $t$= 2 ps and $t$= 9 ps respectively,
with those from DFT-based calculations[4] in Fig 8. Note that our calculation
underestimates the Fermi level for $T_{e}$ = 10200 K ($t=2$ ps) by a few eV,
leading to the observed shift in the XANES.

Figure 8: Time-resolved XANES for 70 nm copper at different delay-times $t=$2
ps and 9 ps. The experimental measurement (red) is averaged over 150
snapshots. The ambient condition measurement is averaged over 70 ps. Our
calculations (blue) for different times are computed at $T_{e}=$ 10200 K and
6000 K. The DFT results[4] (green) are calculated using the Quantum Espresso
code at the same temperatures.

Figure 9: $\textnormal{L}_{3}$-edge absorption for titanium (Top) where the
structure reflects that of the unfilled d-bands, and gold (Bottom) where the
d-bands are nearly filled. The difference ${\Delta}$XAS between the XAS at
finite $T$ and zero temperature is also shown.

###  IV.3 Transition Metals: Ti and Au

As examples of other FT $L_{3}$-edge XAS calculations, we present FT
calculations for a an early transition metal (titanium), where the $d$-bands
are partially filled, and for a late transition metals (gold) where the
$d$-bands are full. Fig. 9. shows the temperature dependence of the L-edges of
these materials. The XANES of Ti is blue-shifted with increasing temperature
because the chemical potential is located in the middle of the $d$-band and
the density of states above the chemical potential is higher. Also, the
broadening of the Fermi-Dirac distribution includes a wider energy range of
$d$-states in the transition, which affects the onset of the pre-edge. In
contrast, the XANES of gold (Au) is red-shifted, as shown in bottom plot of
Fig. 9. This opposing behavior is due to the higher density of states below
the chemical potential. Again, the onset of the pre-edge structure is due to
the broadening of the Fermi-Dirac distribution and shift of the chemical
potential.

###  IV.4 MgO

Figure 10: Mg K-edge spectra for MgO at temperature T=300 K (blue) and 870 K
(red). The top plot shows the experimental spectra[37], the DFT spectra [37]
and the FEFF quasi-harmonic spectra. The bottom plot shows the spectra
difference with respect to T=300K for the experiment (solid black), FEFF
(solid blue) and DFT (dashed red) are shown.

As a final example, we present results for the Mg K-edge XANES of MgO with FT
lattice behavior calculated using the quasi-harmonic approximation. Fig. 10
shows the average XAS for 30 randomly sampled configurations at equilibrium
temperatures $T_{e}=T_{l}=$ 300K and 870K. The configurations are obtained by
sampling the normal modes calculated from dynamical matrices at the
experimental lattice constants and the given temperatures using Quantum
Espresso code [30, 31, 32], with GGA functional[33] for a 16-atom 2 $\times$ 2
$\times$ 2 supercell constructred from a primitive cell. The electronic
density of state is computed with a 6 $\times$ 6 $\times$ 6 k-point grid with
an energy cutoff of 58 Ry. Next, we use the density-functional perturbation
theory to compute the dynamical matrix calculation at the $\Gamma$-point. The
long range electric fields is accounted for from the calculation of the Born
effective charge.

The FEFF calculations are found to overestimate the screening effect of light
element oxides [34]. Instead of using the ad-hoc $Z+1$ approach to overcome
the strong screening effect of the final state rule, we used the random-phase
approximation (RPA) for the core hole screening in FEFF. The RPA is more
accurate in this case, being similar to a Bethe-Salpeter equation. [35, 36]
Finally, we add an additional energy shift of 2.6 eV to the chemical
potential. As a comparison, we show the experiment and DFT calculation by
Nemausat  _et al._ using the stochastic self-consistent harmonic
approximation[37] in Fig. 10. Our muffin-tin potential model predicts a
smaller excitations between 1310 eV and 1315 eV.

##  V CONCLUSIONS

We have extended the real-space Green’s function theory of XAS [15] to finite
temperatures (FT) up to the warm-dense matter regime $T\approx T_{F}$. Our FT
generalization takes into account both electron temperature and lattice
temperature effects. Briefly, a FT SCF procedure is carried out in the complex
energy plane in terms of the FT one-electron Green’s function. This builds in
the FT exchange-correlation potential, using the KSDT parameterization. The FT
self-energy is also important for XAS, since it accounts for final-state
effects in the spectra including corrections to shifts and broadening. While
dynamic exchange effects can also be included, due to the difficulty
calculating the FT self-energy, a more efficient FT COHSEX approximation has
been used. An important difference from the $T=0$ theory of the XAS is an
account of smearing of the Fermi level due to FT occupation numbers.
Vibrational effects are included in terms of a correlated Debye model for the
fine-structure at low $T$ in XAS or a configurational average for XANES at
high $T$. The FT generalization introduced here is implemented as an extension
of the FEFF codes in a new version FEFF10 [12]. The approach has been tested
against various experiments, and typically yields good quantitative agreement.
We believe these developments may be useful in interpretation of many
experiments, e.g., for studies non-equilibrium behavior, extreme conditions,
and shocked conditions. The approach can also be used to differentiate between
lattice and electron temperature effects.

###### Acknowledgements.

We thank R. Albers, B. Fultz, D. Prendergast, G. Seidler, S. Trickey, F. Vila,
and H. Wende for comments and suggestions. This work is supported by DOE BES
Grant DE-FG02-97ER45623. The development of FEFF10 carried out within the
Theory Institute for Materials and Energy Spectroscopies (TIMES) at SLAC, and
is supported by the U.S. DOE, Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering, under contract DE-AC02-76SF00515.

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# A calculation of the Weyl anomaly for 6D Conformal Higher Spins

R Aros1 F Bugini2 and D E Diaz3 1 Departamento de Ciencias Físicas,
Universidad Andres Bello, Sazié 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 Físicas, Universidad Andres Bello, Autopista Concepción-Talcahuano
7100, Talcahuano, Chile<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we continue the study of the one-loop partition function for
higher derivative conformal higher spin (CHS) fields in six dimensions and its
holographic counterpart given by massless higher spin Fronsdal fields in seven
dimensions.

In going beyond the conformal class of the boundary round 6-sphere, we start
by considering a Ricci-flat, but not conformally flat, boundary and the
corresponding Poincaré-Einstein space-filling metric. Here we are able to
match the UV logarithmic divergence of the boundary with the IR logarithmic
divergence of the bulk, very much like in the known 4D/5D setting, under the
assumptions of factorization of the higher derivative CHS kinetic operator and
WKB-exactness of the heat kernel of the dual bulk field. A key technical
ingredient in this construction is the determination of the fourth heat kernel
coefficient $b_{6}$ for Lichnerowicz Laplacians on both 6D and 7D Einstein
manifolds. These results allow to obtain, in addition to the already known
type-A Weyl anomaly, two of the three independent type-B anomaly coefficients
in terms of the third, say $c_{3}$ for instance.

In order to gain access to $c_{3}$, and thus determine the four central
charges independently, we further consider a generic non Ricci-flat Einstein
boundary. However, in this case we find a mismatch between boundary and bulk
computations for spins higher than two. We close by discussing the nature of
this discrepancy and perspectives for a possible amendment.

##  1 Introduction

The AdS/CFT correspondence [1, 2, 3] has evolved over the years into a central
subject of interest in theoretical physics. While many of the succeeding
efforts have been devoted to applications in condensed matter, nuclear and
particle physics, it seems fair to say that quantum gravity aspects of the
duality are still largely unexplored. The most studied regime so far considers
tree level bulk gravity and leading $1/N$ on the boundary. Genuine quantum
gravity effects come out in this regime as a perturbative expansion in terms
of loop Witten diagrams, which have proven quite difficult to compute.
Nonetheless, there has been recent progress in computing one-loop Witten
diagrams (see e.g. [4, 5, 6]).

There is yet another instance that probes one-loop quantum effects on both
bulk and boundary theories in terms of partition functions. It consists of a
class of bulk-boundary fluctuation determinant relation of the type [7, 8],

$\frac{Z^{{}^{1-loop,-}}_{{}_{bulk}}}{Z^{{}^{1-loop,+}}_{{}_{bulk}}}\,=\,Z^{{}^{1-loop}}_{{}_{bndry}}~{}.$
(1)

The bulk side contemplates the ratio of the functional determinants of the
kinetic operator of the bulk field computed with standard and alternate
boundary conditions. The boundary side involves the functional determinant of
the kinetic operator of the induced field. This kind of holographic formula
was obtained via a rather circuitous route within AdS/CFT correspondence, in
connection with a class of RG flows triggered by double-trace deformations of
the CFT [7, 8, 9]. Full match for a massive scalar in Euclidean AdS bulk was
first shown in [10], and by now there are plenty of extensions: an incomplete
list includes fields with nonzero spin (Dirac, MHS, etc.) and quotients of AdS
space (thermal AdS, BTZ, singular AdS, etc.) [11, 12].

The aim in the present work is to proceed with the analysis of the holographic
formula for boundary conformal higher spin fields in six dimensions and to
extend the dictionary already established in the conformally flat situation
[13]. In particular, we explore Ricci-flat but non-conformally flat boundaries
as was done in 4D [14] for CHS and also in 6D for GJMS [12] and for the Weyl
graviton[15]. In doing so, a crucial technical breakthrough is achieved by the
computation of the heat coefficient $b_{6}$ for Lichnerowicz Laplacians on an
Einstein manifold background. We find out a remarkable general combinatorial
structure of the coefficient valid in any dimension. More specifically, the 6D
results perfectly agree with those obtained via the group theory method in
[16] for totally symmetric rank-s tensors in 6D. We then examine the
consequences of the extension of the above mapping to a generic Einstein
boundary manifold and discuss the nature of the mismatch we find as well as
suggestions for possible amendments.

##  2 Preliminaries: conformal flatness

In general, the extension of conformal higher spins to curved backgrounds can
be quite challenging. However, on conformally flat Einstein backgrounds (e.g.
(A)dS, $S$, $H$) the corresponding Weyl-covariant 2s+2-derivative kinetic
operator factorizes into products of standard $2^{\text{nd}}$-derivative
operators. For the 6D sphere this was explicitly exploited in [17] to compute
the UV-log divergence of the one-loop partition function, and thus the type-A
anomaly coefficient, using $\zeta$-function regularization. According to the
holographic formula, the counterpart in the bulk is given by the following
ratio of functional determinants on AdS7

$Z^{{}^{1-loop}}_{{}_{bulk}}=\left[\frac{\det_{\bot\top}\left\\{\hat{\Delta}_{L}^{(s)}+2(s-1)(s+4)\right\\}}{\det_{\bot\top}\left\\{\hat{\Delta}_{L}^{(s-1)}+2(s-1)(s+4)\right\\}}\right]^{-1/2}.$
(2)

The IR-log divergence in this conformally flat situation was first derived in
[18], using the spectral $\zeta-$function to evaluate the trace of the heat
kernel at coincident points, for the whole family of bulk higher spin gauge
fields. This completely agrees with the UV-log divergence of the boundary
computation [17].

To begin our discussion, we will first reproduce the holographic derivation
based on Eq.(2) above in a slightly different way by exploiting the WKB-
exactness of the diagonal heat kernel instead. The heat kernels for totally
symmetric transverse-traceless rank-s tensors in hyperbolic space have long
been known [19, 20, 21]. A remarkable feature, referred to as WKB exactness,
is that after factorization of the exponential of a multiple of the Ricci
scalar 111We set the radius of the hyperbolic space to 1, so that the bulk
Ricci scalar is simply $\hat{R}=-42$., the trace only has finite many terms so
that the combined physical (rank s) and ghost (rank s -1) contributions are
given by the following proper-time integrals (in the standard route to the
log-dets.)

$\displaystyle\int_{0}^{\infty}\frac{dt}{t}\left\\{\mbox{tr}_{{}_{\bot\top}}\,e^{-\left\\{\hat{\Delta}_{L}^{(s)}+2(s-1)(s+4)\right\\}\,t}\,-\,\mbox{tr}_{{}_{\bot\top}}\,e^{-\left\\{\hat{\Delta}_{L}^{(s-1)}+2(s-1)(s+4)\right\\}\,t}\,\right\\}$
(3) $\displaystyle\sim$
$\displaystyle\int_{0}^{\infty}\frac{dt}{t^{9/2}}\left\\{e^{-(s+1)^{2}t}g(s)\left[1+\frac{2t}{5}\left(s^{2}+4s+5\right)+\frac{4t^{2}}{15}(s+2)^{2}\right]\right.$
$\displaystyle\kern
40.00006pt-\left.e^{-(s+2)^{2}t}g(s-1)\left[1+\frac{2t}{5}\left(s^{2}+2s+2\right)+\frac{4t^{2}}{15}(s+1)^{2}\right]\right\\}~{},$

where

$g(s)=\displaystyle{\frac{(s+1)(s+2)^{2}(s+3)}{12}}$ (4)

denotes the number of components of the transverse-traceless field of spin
$s$.

After evaluation of the proper-time integrals in terms of Gamma functions, we
obtain the numerical factor corresponding to the one-loop effective Lagrangian
on hyperbolic space that accompanies the volume, following the prescription in
[22]. The volume anomaly is given by the Q-curvature so that we can directly
read off the type-A Weyl anomaly coefficient, it coincides with the result
obtained from the boundary computation on the round six-sphere, written in
terms of the number of dynamical degrees of the rank-s CHS in 6D
$\nu_{s}=\frac{(s+1)^{2}(s+2)^{2}}{4}$,

$a\,={\frac{\nu_{s}}{360\cdot
7!}}\left(88\,{\nu}_{s}^{3/2}-110\,\nu_{s}-4\,{\nu}_{s}^{1/2}+1\right)~{}.$
(5)

##  3 Ricci flat 6D boundary

In this section we explore deviations from conformal flatness. To proceed
with, we first consider Ricci-flat boundary metrics and we assume that the
factorization of the higher-derivative kinetic operator into Laplacians still
holds with a Lichnerowicz-type coupling to the Weyl tensor, thus the one-loop
boundary partition function becomes

$Z^{{}^{1-loop}}_{{}_{bndry}}=\left[\frac{\left(\det_{\top}\left\\{{\Delta}_{L}^{(s)}\right\\}\right)^{s+1}}{\left(\det_{\top}\left\\{{\Delta}_{L}^{(s-1)}\right\\}\right)^{s+2}}\right]^{-1/2}$
(6)

In order to work out the UV log-divergence, the relevant heat coefficient can
be read off from Eq.(B13) in [16] for the $(0,s,0)$ representation, where the
only curvature invariants that come into play due to Ricci-flatness are
$A_{5}$, $A_{16}$ and $A_{17}$. Alternatively, the very same coefficient can
be obtained by our explicit results for the totally symmetric rank-s tensor
when the generic dimension $n$ is set to 6 and the trace components are
subtracted, that is

$b_{6,s}^{\top}\Big{|}_{\text{Ricci-
flat}}=\left(b_{6,s}-b_{6,s-2}\right)\Big{|}_{\text{Ricci-flat}}$ (7)

and

$b_{6,s-1}^{\top}\Big{|}_{\text{Ricci-
flat}}=\left(b_{6,s-1}-b_{6,s-3}\right)\Big{|}_{\text{Ricci-flat}}$ (8)

with

$\displaystyle 7!b_{6},s\Big{|}_{\text{Ricci-flat}}$
$\displaystyle=A_{5}\left\\{-3\cdot\binom{s+5}{5}+56\cdot\binom{s+6}{7}-1260\cdot\binom{s+7}{9}\right\\}$
(9)
$\displaystyle\kern-55.00008pt+A_{16}\left\\{\frac{44}{9}\cdot\binom{s+5}{5}-84\cdot\binom{s+6}{7}+420\cdot\binom{s+7}{9}-23520\cdot\binom{s+8}{11}\right\\}$
$\displaystyle\kern-55.00008pt+A_{17}\left\\{\frac{80}{9}\cdot\binom{s+5}{5}-168\cdot\binom{s+6}{7}+6720\cdot\binom{s+7}{9}+6720\cdot\binom{s+8}{11}\right\\}~{}.$

The accumulated heat coefficients for the physical and ghost fields are simply
given by $(s+1)\cdot b_{6,s}^{\top}$ and $(s+2)\cdot b_{6,s-1}^{\top}$,
respectively, with

$\displaystyle b_{6,s}^{\top}\Big{|}_{\text{Ricci-flat}}$
$\displaystyle=\,g(s){\frac{45\,{s}^{4}+360\,{s}^{3}+383\,{s}^{2}-1348\,s+180}{7!\;60}}\,A_{5}$
(10)
$\displaystyle\kern-35.00005pt-g(s){\frac{28\,{s}^{6}+336\,{s}^{5}+823\,{s}^{4}-2376\,{s}^{3}-5927\,{s}^{2}+9636\,s-880}{7!\;180}}A_{16}$
$\displaystyle\kern-35.00005pt+2g(s){\frac{{s}^{6}+12\,{s}^{5}+121\,{s}^{4}+648\,{s}^{3}+652\,{s}^{2}-2064\,s+200}{7!\;45}}A_{17}.$

In all, the UV log-divergence is essentially given by

$\displaystyle(s+1)b_{6,s}^{\top}-(s+2)b_{6,s-1}^{\top}\Big{|}_{\text{Ricci-
flat}}$ $\displaystyle=-\frac{\nu_{s}}{9\cdot
7!}\left(63\,\nu_{s}-308\,\nu^{1/2}_{s}+272\right)A_{5}$ (11)
$\displaystyle\kern-55.00008pt-\frac{\nu_{s}}{45\cdot
7!}\left(168\,{\nu_{s}}^{3/2}-1785\,\nu_{s}+5376\,{\nu}_{s}^{1/2}-3979\right)A_{16}$
$\displaystyle\kern-55.00008pt+\frac{\nu_{s}}{45\cdot
7!}\left(48\,{\nu_{s}}^{3/2}+1200\,\nu_{s}-6264\,{\nu}_{s}^{1/2}+5416\right)A_{17}.$

Now, the structure of the UV-log divergences of any 6D Weyl invariant action
is dictated by the trace (or Weyl or conformal) anomaly (see e.g. [23]):

${\mathcal{A}}\,=\,-{a}\,E_{6}\,+\,{c_{1}}\,I_{1}\,+\,{c_{2}}\,I_{2}\,+\,{c_{3}}\,I_{3}$
(12)

The restriction to a Ricci-flat, but not conformally flat, background forces
two linear relations between the four terms of the anomaly basis, namely,
$E_{6}=64I_{1}+32I_{2}$ and $I_{3}=4I_{1}-I_{2}$, so that there are only two
independent terms in the anomaly, say

${\mathcal{A}}\,=\,[-64a+c_{1}+4c_{3}]\,I_{1}\,+\,[-32a+c_{2}-c_{3}]\,I_{2}.$
(13)

The dictionary is simple in the boundary Ricci-flat case (see e.g. table 1 of
reference [15]) $A_{5}=-I_{3}=I_{2}-4I_{1}$, $A_{16}=I_{2}$ and
$A_{17}=-I_{1}$, therefore we obtain for the UV log-divergence of the CHS

$\displaystyle{\mathcal{A}}$ $\displaystyle=$
$\displaystyle-\frac{\nu_{s}}{45\cdot
7!}\left(48\,{\nu}_{s}^{3/2}-60\,\nu_{s}-104\,{\nu}_{s}^{1/2}+24\right)I_{1}$
$\displaystyle-$ $\displaystyle\frac{\nu_{s}}{45\cdot
7!}\left(168\,{\nu_{s}}^{3/2}-1470\,\nu_{s}+3836\,{\nu}^{1/2}_{s}-2619\right)I_{2}$

Finally, the new information on the central charges amounts to

$\displaystyle c_{1}+4c_{3}$ $\displaystyle=$
$\displaystyle\frac{\nu_{s}}{45\cdot
7!}\left(656\,{\nu}_{s}^{3/2}-820\,\nu_{s}+72\,{\nu}^{1/2}_{s}+32\right)$ (15)

and

$\displaystyle\qquad\quad c_{2}-c_{3}$ $\displaystyle=$
$\displaystyle\frac{\nu_{s}}{45\cdot
7!}\left(184\,{\nu}_{s}^{3/2}+1030\,\nu_{s}-3852\,{\nu}_{s}^{1/2}+2623\right).$
(16)

As it can be appreciated, the information on the type-B Weyl anomaly
coefficients $c_{1},\,c_{2}$ and $c_{3}$ is partial; essentially $c_{3}$
remains undetermined. It is worth noticing that $c_{3}$ is proportional to the
$C_{T}$ coefficient of the two-point function of the stress-energy tensor,
$C_{T}=3024\,c_{3}$. Alternatively, in the convention of [16]
$c=\frac{c_{2}-c_{3}}{32}$,
$c^{\prime\prime}=\frac{c_{1}-2c_{2}+6c_{3}}{192}=\frac{c_{1}+4c_{3}}{192}-\frac{c_{2}-c_{3}}{96}$,$c^{\prime}=\frac{c_{1}-4c_{2}}{192}=\frac{c_{1}+4c_{3}}{192}-\frac{c_{2}-c_{3}}{48}-\frac{c_{3}}{24}$
it is clear that $c$ and $c^{\prime\prime}$ have been fully determined,
whereas $c^{\prime}$ remains unknown on Ricci-flat spaces.

##  4 Bulk Poincaré-Einstein

Now we will consider the holographic counterpart, consisting in a Poincaré-
Einstein metric with a Ricci-flat representative in the family of conformal
boundary metrics. The bulk partition function contains the physical and the
ghost functional determinants

$Z^{{}^{1-loop}}_{{}_{bulk}}=\left[\frac{\det_{\bot\top}\left\\{\hat{\Delta}_{L}^{(s)}+2(s-1)(s+4)\right\\}}{\det_{\bot\top}\left\\{\hat{\Delta}_{L}^{(s-1)}+2(s-1)(s+4)\right\\}}\right]^{-1/2}$
(17)

To compute holographically the type-B central charges, it suffices to consider
the Weyl content of the $\hat{b}_{6}$ anomaly coefficient when restricting to
boundary Ricci-flat metrics. The Weyl content of the heat coefficient of the
Lichnerowicz Laplacian can be read off from the contribution of the three
curvature invariants $\hat{A}_{5},\hat{A}_{16}$ and $\hat{A}_{17}$. As already
pointed out, they simply follow from extrapolation of the 6D results when
written in terms of combinatorial coefficients

$\displaystyle 7!\cdot\hat{b}_{6},s\big{|}_{\text{Ricci flat}}$
$\displaystyle=\hat{A}_{5}\left\\{-3\cdot\binom{s+6}{6}+56\cdot\binom{s+7}{8}-1260\cdot\binom{s+8}{10}\right\\}$
(18)
$\displaystyle\kern-55.00008pt+\hat{A}_{16}\left\\{\frac{44}{9}\cdot\binom{s+6}{6}-84\cdot\binom{s+7}{8}+420\cdot\binom{s+8}{10}-23520\cdot\binom{s+9}{12}\right\\}$
$\displaystyle\kern-55.00008pt+\hat{A}_{17}\left\\{\frac{80}{9}\cdot\binom{s+6}{6}-168\cdot\binom{s+7}{8}+6720\cdot\binom{s+8}{10}+6720\cdot\binom{s+9}{12}\right\\}~{}.$

Assuming now that the WKB-exactness in the conformally flat case extends to
the Weyl terms in the present Poincaré-Einstein, we end up with the proper
time integrals

$\displaystyle\int_{0}^{\infty}\frac{dt}{t}\left\\{\mbox{tr}_{{}_{\bot\top}}\,e^{-\left\\{\hat{\Delta}_{L}^{(s)}+2(s-1)(s+4)\right\\}\,t}\,-\,\mbox{tr}_{{}_{\bot\top}}\,e^{-\left\\{\hat{\Delta}_{L}^{(s-1)}+2(s-1)(s+4)\right\\}\,t}\,\right\\}$
(19) $\displaystyle\sim$
$\displaystyle\int_{0}^{\infty}\frac{dt}{t^{9/2}}\left\\{e^{-(s+1)^{2}t}\left[t^{3}\hat{b}_{6,s}^{\bot\top}+\ldots\right]-e^{-(s+2)^{2}t}\left[t^{3}\hat{b}_{6,s-1}^{\bot\top}+\ldots\right]\right\\},$

By making use of the holographic prescription (see e.g. table 2 of reference
[15]), we find the contribution to the boundary Weyl anomaly in terms of the
two independent boundary curvature invariants, say $I_{1}$ and $I_{2}$, that
remain. Let us explain in details how this is done. First, one has to notice
that the bulk invariant $\hat{A}_{12}=-42\hat{W}^{2}$ descends to a multiple
of the boundary $R\,W^{2}$ that vanishes on the Ricci-flat boundary; this
restricts the basis of bulk invariants since now the combination corresponding
to the above $-4\hat{W}^{{}^{\prime}\,3}+\hat{W}^{3}-\hat{\Phi}_{7}$ does not
produce any contribution to the holographic Weyl anomaly. By focusing only on
the cubic Weyl terms, for which the holographic prescription is trivially
$\hat{W}^{{}^{\prime}\,3}\rightarrow I_{1}$ and $\hat{W}^{3}\rightarrow
I_{2}$, we end up with the same prescription for the hatted quantities as in
the boundary, namely $\hat{A}_{17}\rightarrow-I_{1}$, $\hat{A}_{16}\rightarrow
I_{2}$ and
$\hat{A}_{5}\sim\hat{\Phi}_{7}\sim-4\hat{W}^{{}^{\prime}\,3}+\hat{W}^{3}\rightarrow-4I_{1}+I_{2}$.
The answer completely agrees with what we obtained on the boundary under the
assumption of factorization of the higher-derivative kinetic operator. Thus we
see that they go hand by hand, WKB exactness of the bulk heat kernel and
boundary factorization. The obvious reason for the 6D/7D matching can be
traced back to the agreement between the heat coefficients $b_{6,s}^{\top}$
and $\hat{b}_{6,s}^{\bot\top}$ when written in terms of the boundary and bulk
invariants, respectively, when restricted to a Ricci-flat boundary 222This
identity lies at the heart of the functional Schrödinger method, see e.g.
[16].

##  5 Generic Einstein 6D boundary

In order to gain access to the remaining $c_{3}$ coefficient it is necessary
to depart from Ricci-flat and conformally flat boundaries. In this section we
will explore the natural extension to a generic Einstein 6D boundary and, for
practical purposes, we will assume that factorization into Lichnerowicz-type
Laplacians still holds on. Therefore, the boundary partition function becomes
[17]

$Z^{{}^{1-loop}}_{{}_{bndry}}=\left[\frac{\displaystyle{\prod_{k^{\prime}=-1}^{s-1}}\det_{\bot\top}\left\\{{\Delta}_{L}^{(s)}-\frac{R}{30}[s(s+3)+(k^{\prime}-1)(k^{\prime}+4)]\right\\}}{\displaystyle{\prod_{k=0}^{s-1}}\det_{\bot\top}\left\\{{\Delta}_{L}^{(k)}-\frac{R}{30}[k(k+3)+(s-1)(s+4)]\right\\}}\right]^{-1/2}$
(20)

When going from a Ricci-flat boundary to a generic Einstein one, in addition
to the curvature invariants $A_{5}$, $A_{16}$ and $A_{17}$, two new ones
involving the Weyl tensor come into play, namely $A_{12}$ and $A_{15}$. Since
our present interest is the computation of the remaining $c_{3}$ coefficient,
in what follows we will omit the pure-Ricci curvature invariants
$A_{10},A_{11},A_{13}$ and $A_{14}$ that reduce to multiples of the cubed
Ricci scalar $R^{3}$ and thus only contribute to the $a$ anomaly coefficient.
One possible way to incorporate these two new contributions into $b_{6}$ is by
careful examination of the entire list of traces of the $V^{\prime}s$
coefficients in equation (36), as worked out in [16] (see section B.4
therein), for the $(0,s,0)$ representation. Remarkably, the overall
contributions to $A_{12}$ and $A_{15}$ obtained for $b_{6,s}^{\top}$ in this
way exactly match our explicit results for the totally symmetric rank-$s$
tensor in table 1 when the generic dimension $n$ is set to 6 and the trace
components are subtracted as

$\left.b_{6,s}^{\top}\right|_{\text{Einstein}}=\left.\left(b_{6,s}-b_{6,s-2}\right)\right|_{\text{Einstein}}$
(21)

where $\left.b_{6,s}\right|_{\text{Einstein}}$, now improved by the presence
of $A_{12}$ and $A_{15}$, is given by

$\displaystyle 7!\left.b_{6,s}\right|_{\text{Einstein}}$
$\displaystyle=A_{5}\left\\{-3\cdot\binom{s+5}{5}+56\cdot\binom{s+6}{7}-1260\cdot\binom{s+7}{9}\right\\}$
(22)
$\displaystyle+A_{12}\left\\{\frac{14}{3}\cdot\binom{s+5}{5}-98\cdot\binom{s+6}{7}+1680\cdot\binom{s+7}{9}-7560\cdot\binom{s+8}{11}\right\\}$
$\displaystyle+A_{15}\left\\{-\frac{16}{3}\cdot\binom{s+5}{5}+476\cdot\binom{s+6}{7}-13440\cdot\binom{s+7}{9}-60480\cdot\binom{s+8}{11}\right\\}$
$\displaystyle+A_{16}\left\\{\frac{44}{9}\cdot\binom{s+5}{5}-84\cdot\binom{s+6}{7}+420\cdot\binom{s+7}{9}-23520\cdot\binom{s+8}{11}\right\\}$
$\displaystyle+A_{17}\left\\{\frac{80}{9}\cdot\binom{s+5}{5}-168\cdot\binom{s+6}{7}+6720\cdot\binom{s+7}{9}+6720\cdot\binom{s+8}{11}\right\\}~{}.$

The Weyl content in the heat coefficient for the Lichnerowicz Laplacian on
traceless totally symmetric rank-s fields is now given by

$\displaystyle b_{6,s}^{\top}$
$\displaystyle=\,g(s){\frac{45\,{s}^{4}+360\,{s}^{3}+383\,{s}^{2}-1348\,s+180}{7!\;60}}\,A_{5}$
(23)
$\displaystyle\kern-35.00005pt-g(s){\frac{3\,{s}^{6}+36\,{s}^{5}+33\,{s}^{4}-696\,{s}^{3}-1100\,{s}^{2}+2704\,s-280}{7!\;60}}A_{12}$
$\displaystyle\kern-35.00005pt-2g(s){\frac{3\,{s}^{6}+36\,{s}^{5}+153\,{s}^{4}+264\,{s}^{3}-95\,{s}^{2}-956\,s+40}{7!\,45}}A_{15}$
$\displaystyle\kern-35.00005pt-g(s){\frac{28\,{s}^{6}+336\,{s}^{5}+823\,{s}^{4}-2376\,{s}^{3}-5927\,{s}^{2}+9636\,s-880}{7!\;180}}A_{16}$
$\displaystyle\kern-35.00005pt+2g(s){\frac{{s}^{6}+12\,{s}^{5}+121\,{s}^{4}+648\,{s}^{3}+652\,{s}^{2}-2064\,s+200}{7!\;45}}A_{17}.$

Notice that the shifts in the Lichnerowicz Laplacians entering the partition
function, Eq.(20), modify the accumulated heat coefficient for the physical
and ghost fields by terms of the form $R\cdot b^{\top}_{4,s}$. This modifies
the overall coefficient of $A_{12}$ as

$R\cdot
b^{\top}_{4,s}=g(s)\frac{9\,{s}^{4}+72\,{s}^{3}+71\,{s}^{2}-292\,s+56}{7!\cdot
24}\,A_{12}.$ (24)

Once the shifts are properly taken into account, it only remains to keep track
on the dependence on $I_{3}$ for each curvature invariant. See the extracts of
the dictionary in table 1 of reference [15]

$A_{5}\sim\frac{5}{3}I_{3},\,A_{12}\sim-5I_{3},\,A_{15}\sim-\frac{5}{6}I_{3},\,A_{16}\sim-
I_{3}\;\mbox{and}\;A_{17}\sim\frac{1}{4}I_{3}.$ (25)

Since the structure of the trace anomaly on the generic Einstein 6D manifold
gets rearranged in a pure-Ricci term, that is $R^{3}$ which is essentially the
6D Q-curvature, and shifted coefficients for the three Weyl invariants

${\mathcal{A}}\,=\,-16{a}\,R^{3}\,+\,{(c_{1}-96\,a)}\,I_{1}\,+\,{(c_{2}-24\,a)}\,I_{2}\,+\,{(c_{3}+8\,a)}\,I_{3},$
(26)

the overall coefficient of $I_{3}$ grants access to the sought-after $c_{3}$
charge

$c_{3}=-\frac{\nu_{s}}{45\cdot
7!}\left(76\,{\nu}_{s}^{3/2}+745\,\nu_{s}-3548\,{\nu}^{1/2}_{s}+2637\right).$
(27)

##  6 Bulk Poincaré-Einstein with Einstein boundary

For the corresponding space-filling metric, whose boundary has been discussed
in the previous section, we choose the Poincaré-Einstein metric

$\hat{g}_{{}_{\text{PE/E}}}=\displaystyle{\frac{dx^{2}+(1-\lambda
x^{2})^{2}g_{{}_{\text{E}}}}{x^{2}}}$ (28)

where $x=0$ defines the boundary, $g_{{}_{\text{E}}}$ is the Einstein metric
on the conformal boundary and $\lambda=R/120$ is a multiple of the
(necessarily constant) Ricci scalar $R$ of the (6-dimensional) boundary
Einstein metric. With that in mind, we examine the bulk partition function
given by [17, 18]

$Z^{{}^{\text{1-loop}}}_{{}_{\text{bulk}}}=\left[\frac{\det_{\bot\top}\left\\{\hat{\Delta}_{L}^{(s)}+2(s-1)(s+4)\right\\}}{\det_{\bot\top}\left\\{\hat{\Delta}_{L}^{(s-1)}+2(s-1)(s+4)\right\\}}\right]^{-1/2}$
(29)

It is worth noticing that, in correspondence with the boundary computation,
there are again two additional bulk curvature invariants $\hat{A}_{12}$ and
$\hat{A}_{15}$ that come into play. Thus, for the unconstrained bulk field we
obtain

$\displaystyle 7!\hat{b}_{6,s}$
$\displaystyle=\hat{A}_{5}\left\\{-3\cdot\binom{s+6}{6}+56\cdot\binom{s+7}{8}-1260\cdot\binom{s+8}{10}\right\\}$
(30)
$\displaystyle+\hat{A}_{12}\left\\{\frac{14}{3}\cdot\binom{s+6}{6}-98\cdot\binom{s+7}{8}+1680\cdot\binom{s+8}{10}-7560\cdot\binom{s+9}{12}\right\\}$
$\displaystyle+\hat{A}_{15}\left\\{-\frac{16}{3}\cdot\binom{s+6}{6}+476\cdot\binom{s+7}{8}-13440\cdot\binom{s+8}{10}-60480\cdot\binom{s+9}{12}\right\\}$
$\displaystyle+\hat{A}_{16}\left\\{\frac{44}{9}\cdot\binom{s+6}{6}-84\cdot\binom{s+7}{8}+420\cdot\binom{s+8}{10}-23520\cdot\binom{s+9}{12}\right\\}$
$\displaystyle+\hat{A}_{17}\left\\{\frac{80}{9}\cdot\binom{s+6}{6}-168\cdot\binom{s+7}{8}+6720\cdot\binom{s+8}{10}+6720\cdot\binom{s+9}{12}\right\\}~{}.$

Furthermore, there is an additional contribution to $\hat{A}_{12}$ coming from

$\hat{R}\cdot\hat{b}_{4,s}=\hat{A}_{12}\left\\{\frac{1}{180}\left(\begin{array}[]{c}s+6\\\
6\end{array}\right)-\frac{1}{12}\left(\begin{array}[]{c}s+7\\\
8\end{array}\right)+\frac{3}{2}\left(\begin{array}[]{c}s+8\\\
10\end{array}\right)\right\\}$ (31)

To determine the Weyl content of the IR-log divergence, under the assumption
of WKB-exactness of the heat kernel for transverse traceless bulk fields, we
end up with the proper-time integrals

$\displaystyle\int_{0}^{\infty}\frac{dt}{t}\left\\{\mbox{tr}_{{}_{\bot\top}}\,e^{-\left\\{\hat{\Delta}_{L}^{(s)}+2(s-1)(s+4)\right\\}\,t}\,-\,\mbox{tr}_{{}_{\bot\top}}\,e^{-\left\\{\hat{\Delta}_{L}^{(s-1)}+2(s-1)(s+4)\right\\}\,t}\,\right\\}$
(32) $\displaystyle\sim$
$\displaystyle\int_{0}^{\infty}\frac{dt}{t^{9/2}}\left\\{e^{-(s+1)^{2}t}\left[t^{2}\hat{b}_{4,s}^{\bot\top}+t^{3}\hat{b}_{6,s}^{\bot\top}+\ldots\right]-e^{-(s+2)^{2}t}\left[t^{2}\hat{b}_{4,s-1}^{\bot\top}+t^{3}\hat{b}_{6,s-1}^{\bot\top}+\ldots\right]\right\\}.$

It suffices to keep track on the dependence of the bulk curvature invariants
on the Fefferman-Graham invariant $\hat{\Phi}_{7}$ if we are only interested
in the $c_{3}$ coefficient of the holographic anomaly. Again, from the
dictionary in table 2 of reference [15], the contribution to $\hat{\Phi}_{7}$
from every bulk curvature invariant is pinned down

$\hat{A}_{5}\sim
3\hat{\Phi}_{7},\,\hat{A}_{12}\sim-\frac{21}{2}\hat{\Phi}_{7},\,\hat{A}_{15}\sim-\frac{3}{2}\hat{\Phi}_{7},\,\hat{A}_{16}\sim-\frac{3}{2}\hat{\Phi}_{7},\;\text{
and }\;\hat{A}_{17}\sim\frac{3}{8}\hat{\Phi}_{7}.$ (33)

The prescription to compute the holographic Weyl anomaly maps the bulk
$\hat{\Phi}_{7}$ to the boundary $\Phi_{6}$. Additionally, on the Einstein
boundary $I_{3}=16I_{1}-4I_{2}+3\Phi_{6}$ is satisfied and therefore the
overall coefficient of $\hat{\Phi}_{7}$ is simply three times that of $I_{3}$,
i.e., $3(c_{3}+8\,a)$.

Finally, the results above then yield a holographically derived
$\tilde{c}_{3}$ given by

$\tilde{c}_{3}=-\frac{\nu_{s}}{45\cdot
7!}\left(148\,{\nu}_{s}^{3/2}+25\,\nu_{s}-1604\,{\nu}^{1/2}_{s}+1341\right),$
(34)

which differs from its boundary counterpart in Eq.(27). The difference is
given by

$c_{3}-\tilde{c}_{3}=\frac{8\,\nu_{s}}{5\cdot
7!}\left({\nu}_{s}^{1/2}-1\right)\left({\nu}_{s}^{1/2}-3\right)\left({\nu}_{s}^{1/2}-6\right).$
(35)

It must be noticed that the zeros of the expression above correspond to
$s=0,1$ and $2$, as expected, since for these values the mapping has already
been verified. The enforced agreement for these values of the spin essentially
fixes the polynomial up to an overall factor. For comparison purposes, it
would then be desirable to explore an alternative, third, route 333One
concrete possibility relies on the computation of Rényi entropy coefficients
[24]. to $c_{3}$ or, equivalently, to $C_{T}$.

##  7 Conclusion and prospects

In this work we continued the study of conformal higher spin gauge fields and
their holographic dual given by massless higher spin fields in the bulk. We
considered the boundary to be an Einstein manifold, and therefore Bach-flat,
and a dual Poincaré-Einstein bulk metric. It was also assumed a minimal
departure from the conformally flat setting by allowing a Lichnerowicz-type
coupling with both boundary and bulk Weyl tensors.

For totally symmetric rank-$s$ tensors we computed the $b_{6,s}$ heat
coefficient for the Lichnerowicz Laplacian on a generic Einstein background in
any dimension. With this key ingredient, we were able to probe the holographic
formula for one-loop partition functions under the assumptions of
factorization of the CHS higher-derivative kinetic operator and WKB-exactness
of the heat kernel for the MHS bulk field. For a Ricci-flat boundary, we were
able to show the exact matching between the UV-log and IR-log divergences of
the boundary and bulk partition functions, respectively. In addition, we were
also able to obtain partial information on the type-B anomaly coefficients:
determined $c_{1}$ and $c_{2}$ in terms of $c_{3}$, but left the latter
undetermined. It would be interesting to extend these results to include half-
integer spins and to check whether the combinatorial structure of the traces
in $b_{6}$, obtained by the group theory method of [16], admits a ready
extrapolation to generic dimensions.

Considering a generic Einstein boundary, beyond Ricci-flatness, grants access
to $c_{3}$. In this case, however, we found a mismatch between the boundary
and bulk computations. This mismatch does not affect the functional dependence
of $c_{1}$ and $c_{2}$ on $c_{3}$. We must point out that the validity of our
results on the boundary relies on the factorization conjecture of the CHS
kinetic operator, which has been questioned [25]. In particular, it has been
argued that the factorization is obstructed at first order in the background
curvature.

However, as in the 4D case, whether the obstruction terms modify the type-B
Weyl anomaly coefficients $c^{\prime}s$ remains an open issue. Moreover, it
must be stressed that the argument in [25] was under the assumption that the
CHS kinetic operator is diagonal and gauge invariant for each $s$
independently, which seems to fail in general beyond the small curvature
expansion [26]. For instance, in [27] it was shown that for the spin $3$
kinetic term, the gauge invariance can be attained by switching on a coupling
to a conformal spin$-1$ field. This produces a shift in the anomaly
$c$-coefficients. The vanishing of the mismatch on Ricci-flat backgrounds
might be attributed to the fact that terms obstructing the factorization have
no bearing on the log-divergence, and hence, on the Weyl anomaly.

In all, even though the program of constructing CHS gauge fields in a curve
background is far from being settled, even for the spin-3, we expect that once
the mixing issue is resolved a proper holographic picture will come up for
free fields and our present results, although somewhat preliminary, will prove
useful in that endeavour. Let us close by mentioning an interesting approach
to bypass the problem of factorization and further study the mixing of CHS
that has been recently proposed in [28]. This shifts the focus to maximal
depth CHS fields and decreases the order of the kinetic operator. The bulk
dual clearly corresponds to a (partially-)massive higher-spin field, but the
details of the holographic picture must yet be unraveled. Constraints on the
gravitational background are to be expected according to the analysis in [29]
(or, more recently, [30]), but then again this analysis overlooked the
possibility of a mixing with lower spins in order to maintain Weyl invariance.
It would be interesting to investigate these alternatives in the future.

We would like to thank P. Sundell for useful conversations.

##  Appendix A Traces

Here we collect the curvature invariants that contribute to the Weyl content
of the $b_{6}$ heat coefficient for $-\nabla^{2}-E$ when acting on
unconstrained tensor fields, as given by [31, 23]. When restricted to an
Einstein metric, the only contributions come from the invariants
$A_{5},A_{12},A_{15},A_{16}$ and $A_{17}$ ($A_{9}=-A_{5}$ modulo a trivial
total derivative, the same holds for the traces $V_{3}=-V_{1}$ and
$V_{10}=-V_{9}$)

$\displaystyle{b}_{6}\\{-{\nabla}^{2}-E\\}\,=$
$\displaystyle\,\frac{1}{(4\pi)^{3}7!}\,\mbox{tr}_{V}\bigg{\\{}-3A_{5}+\frac{14}{3}A_{12}-\frac{16}{3}A_{15}+\frac{44}{9}A_{16}+\frac{80}{9}A_{17}$
(36) $\displaystyle+14\bigg{(}-4V_{1}-12V_{4}+6V_{5}-4V_{6}+5V_{7}$
$\displaystyle-30V_{10}+60V_{11}+30V_{12}+30V_{16}+2V_{20}\bigg{)}\bigg{\\}}$

Endomorphism The index structure of the endomorphism (Lichnerowicz coupling
containing the Weyl tensor) is as follows (see e.g. [13, 29])

$E=s(s-1)R^{\rho_{1}\,\rho_{2}}_{~{}\nu_{1}~{}\nu_{2}}\delta_{\rho_{1}}^{(\mu_{1}}\delta_{\rho_{2}}^{\mu_{2}}\delta_{\nu_{3}}^{\mu_{3}}\cdots\delta_{\nu_{s}}^{\mu_{s})}-sR^{\rho_{1}}_{~{}\nu_{1}}\delta_{\rho_{1}}^{(\mu_{1}}\delta_{\nu_{2}}^{\mu_{2}}\delta_{\nu_{3}}^{\mu_{3}}\cdots\delta_{\nu_{s}}^{\mu_{s})}\\\
$ (37)

Connection curvature The curvature of the spin connection for the rank$-s$
totally symmetric tensor is given by (see e.g.[13])

$F=\frac{1}{2}R_{\mu\nu}^{\;\;\;ab}\,\Sigma_{ab}$ (38)

with $\Sigma_{ab}$ realizing the Lorentz algebra

$\displaystyle[\Sigma_{ab},\Sigma_{cd}]=-\eta_{ac}\Sigma_{bd}+\eta_{ad}\Sigma_{bc}+\eta_{bc}\Sigma_{ad}-\eta_{bd}\Sigma_{ac}$
(39)

The explicit index structure of the curvature 444This corrects a misprint in
equation (A17) in [14]., generalizing spins one and two, is given by

$(\Sigma_{ab})_{c_{1}\cdots c_{s}}^{d_{1}\cdots
d_{s}}=2s\,\delta_{[a}^{(d_{1}}\eta_{b](c_{1}}\delta_{c_{2}}^{d_{2}}\cdots\delta_{c_{s})}^{d_{s})}$
(40)

$\mbox{tr}_{V}V_{1}$ | $=\mbox{tr}_{V}|{\nabla}{F}|^{2}$ | $=\displaystyle{-\binom{n+s}{n+1}\cdot A_{5}}$
---|---|---
$\mbox{tr}_{V}V_{4}$ | $=\mbox{tr}_{V}{F}^{3}$ | $=\phantom{-}\displaystyle{\binom{n+s}{n+1}\cdot A_{17}}$
$\mbox{tr}_{V}V_{5}$ | $=\mbox{tr}_{V}\text{Riem }{F}^{2}$ | $=\displaystyle{-\binom{n+s}{n+1}\cdot A_{16}}$
$\mbox{tr}_{V}V_{6}$ | $=\mbox{tr}_{V}\text{Ric }{F}^{2}$ | $=\displaystyle{-\binom{n+s}{n+1}\cdot A_{15}}$
$\mbox{tr}_{V}V_{7}$ | $=\mbox{tr}_{V}\text{R }{F}^{2}$ | $=\displaystyle{-\binom{n+s}{n+1}\cdot A_{12}}$
$\mbox{tr}_{V}V_{10}$ | $=\mbox{tr}_{V}|{\nabla}{E}|^{2}$ | $=\displaystyle{3\binom{n+1+s}{n+3}\cdot A_{5}}$
$\mbox{tr}_{V}V_{11}$ | $=\mbox{tr}_{V}{E}^{3}$ | $=\phantom{-}\displaystyle{\left[8\binom{n+2+s}{n+5}+8\binom{n+1+s}{n+3}\right]\cdot A_{17}}$
$\phantom{=}\displaystyle{-\left[28\binom{n+2+s}{n+5}+\binom{n+1+s}{n+3}\right]\cdot
A_{16}}$
$\phantom{=}\displaystyle{-\left[72\binom{n+2+s}{n+5}+18\binom{n+1+s}{n+3}\right]\cdot
A_{15}}$
$\phantom{=}\displaystyle{-9\binom{n+2+s}{n+5}\cdot A_{12}}$
$\mbox{tr}_{V}V_{12}$ | $=\mbox{tr}_{V}{E}{F}^{2}$ | $=\displaystyle{3\binom{n+1+s}{n+3}\cdot A_{16}}\displaystyle{+\left[4\binom{n+1+s}{n+3}+\binom{n+s}{n+1}\right]\cdot A_{15}}$
$\phantom{=}\displaystyle{+\binom{n+1+s}{n+3}\cdot A_{12}}$
$\mbox{tr}_{V}V_{16}$ | $=\mbox{tr}_{V}\text{R }{E}^{2}$ | $=\displaystyle{3\binom{n+1+s}{n+3}\cdot A_{12}}$
$\mbox{tr}_{V}V_{20}$ | $=\mbox{tr}_{V}\text{Riem}^{2}{E}$ | $=\displaystyle{-\binom{n+s}{n+1}\cdot A_{12}}$
Table 1: Traces by explicit contraction for unconstrained totally symmetric
rank-$s$ tensors

Most of the traces reduce to the quadratic ones already computed in a previous
paper [14]. The challenging ones are $V_{4},V_{11}$ and $V_{12}$. We first
contracted the permanents555The relevant identities are Eqs.(A11-A13) in [14],
but for cubic contractions of the Weyl tensor one must subtract traces in the
spectator indices in Eq.(A11). and then used the Cadabra code [32, 33] to
determine the final contractions and read off the coefficients of the
curvature invariants in Table 1.

##  Appendix B Heat coefficient from group theory method

In 6D, the traces entering the heat coefficients were computed via so-called
group theory method [16]. To make contact with our current computation, one
must choose the representation with Dynkin labels $(a,b,c)=(0,s,0)$ that
corresponds to the totally symmetric traceless rank-s tensor.

In general, the dimension of the representation is given by

$\textrm{Dim}_{R}(a,b,c)=\frac{1}{12}(a+1)(b+1)(c+1)(a+b+2)(b+c+2)(a+b+c+3).$
(41)

Using the Weyl character formula it can be shown that the second order Dynkin
index is given by

$I_{2}(a,b,c)=\frac{\textrm{Dim}_{R}}{60}(3a^{2}+2a(2b+c+6)+4b^{2}+4b(c+4)+3c(c+4)).$
(42)

Analogously, the third and fourth order generalizations are given by [34]

$I_{3}(a,b,c)=\frac{\textrm{Dim}_{R}}{120}(a-c)(a+c+2)(a+2b+c+4)$ (43)
$\displaystyle I_{4}(a,b,c)$ $\displaystyle=$
$\displaystyle\frac{\textrm{Dim}_{R}}{3360}(3a^{4}+8a^{3}b+4a^{3}c+24a^{3}+2a^{2}b^{2}$
$\displaystyle+2a^{2}bc+30a^{2}b-4a^{2}c^{2}+6a^{2}c+54a^{2}-2ab^{3}-18ab^{2}c$
$\displaystyle-50ab^{2}+2abc^{2}-28abc-34ab+4ac^{3}+6ac^{2}-2ac+24a$
$\displaystyle-6b^{4}-12b^{3}c-48b^{3}+2b^{2}c^{2}-50b^{2}c-122b^{2}+8bc^{3}+30bc^{2}$
$\displaystyle-34bc-104b+3c^{4}+24c^{3}+54c^{2}+24c)$

These results determine $b_{6,s}^{\top}$ for Ricci-flat geometries, given by

$\displaystyle 7!\cdot b_{6}(R)\Big{|}_{R_{ab}=0}\,=\,$ $\displaystyle
A_{5}\left(\frac{3150I_{2}^{2}}{17\text{Dim}_{R}}+9\text{Dim}_{R}-\frac{1827I_{2}}{17}+\frac{700I_{4}}{17}\right)$
(45)
$\displaystyle\kern-75.0001pt+A_{16}\left(-\frac{9150I_{2}^{3}}{17\text{Dim}_{R}^{2}}+\frac{12900I_{2}^{2}}{17\text{Dim}_{R}}-\frac{560I_{2}I_{4}}{17\text{Dim}_{R}}-\frac{1010I_{3}^{2}}{\text{Dim}_{R}}+\frac{44\text{Dim}_{R}}{9}-\frac{8597I_{2}}{51}+\frac{14140I_{4}}{51}\right)$
$\displaystyle\kern-75.0001pt+A_{17}\left(-\frac{1725I_{2}^{3}}{17\text{Dim}_{R}^{2}}+\frac{10875I_{2}^{2}}{17\text{Dim}_{R}}-\frac{10640I_{2}I_{4}}{17\text{Dim}_{R}}+\frac{2185I_{3}^{2}}{\text{Dim}_{R}}+\frac{80\text{Dim}_{R}}{9}-\frac{17804I_{2}}{51}-\frac{2240I_{4}}{51}\right).$

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