Six-dimensional one-loop divergences in quantum
gravity from the $\mathcal{N}=4$ spinning particle

Fiorenzo Bastianelli${}^{\,a,b}$, Francesco Comberiati${}^{\,a,b}$, Filippo
Fecit${}^{\,a,b}$ and Fabio Ori${}^{\,a}$

aDipartimento di Fisica e Astronomia “Augusto Righi”, Università di Bologna,
via Irnerio 46, I-40126 Bologna, Italy

bINFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy

###### Contents

  1. 1 Introduction
  2. 2 Worldline formalism
    1. 2.1 The worldloop path integral
    2. 2.2 Setting up the perturbative expansion
    3. 2.3 Outline of the computation
    4. 2.4 Seeley-DeWitt coefficients
      1. 2.4.1 Ghost and Graviton separately
  3. 3 Heat kernel method
    1. 3.1 Heat kernel and one-loop effective action
    2. 3.2 Euclidean quantum gravity
  4. 4 On one-loop divergences of quantum gravity
    1. 4.1 Pure gravity in four dimensions
    2. 4.2 Pure gravity in six dimensions
  5. 5 Conclusions
  6. A Basis of invariants on Einstein manifolds
  7. B Worldline computations
    1. B.1 Worldline propagators
    2. B.2 Analisys of perturbative contributions
    3. B.3 Summing up all the pieces
  8. C Heat Kernel computations
  9. D Topological terms in even dimensions

##  1 Introduction

One of the most important and challenging areas of modern theoretical physics
is the search for a consistent quantum theory of gravity. Regardless of its
specific details, any such theory is expected to incorporate and generalize
the principles of general relativity. However, as it is well-known, general
relativity gives rise to a non-renormalizable quantum field theory, in which
diverging terms cannot be absorbed into the parameters of the Einstein-Hilbert
action. This makes the development of a quantum theory of gravity an ongoing
and active area of research, with important implications for our understanding
of the fundamental nature of spacetime.

The investigation of divergences in quantum gravity dates back to the
pioneering work of t’ Hooft and Veltman [1]. They demonstrated that, when
evaluated on-shell and with vanishing cosmological constant, the one-loop
logarithmic divergences in four dimensions were absent. Subsequently, van
Nieuwenhuizen [2] observed that one-loop quantum calculations in six
dimensions share some features with two-loop calculations in four dimensions.
He found that quantum gravity in six dimensions contains a nonvanishing
logarithmic divergence, suggesting that divergences could emerge at two-loops
in four dimensions as well. This prediction was supported by Critchley [3],
who corrected a numerical factor in the six-dimensional term. Eventually,
Goroff and Sagnotti [4] explicitly calculated the two-loop divergence in four
dimensions. They demonstrated that pure quantum gravity is a non-
renormalizable theory at two-loops. This result was later checked and
confirmed by van de Ven [5]. On the other hand, the inclusion of a
cosmological constant gives rise to a one-loop logarithmic divergence already
in four dimensions, as found by Christensen and Duff [6].

Subsequent research has focused on the search and analysis of gravitational
theories with improved ultraviolet behaviour, such as simple supergravity [7,
8] and supergravities with extended supersymmetry, like the $N=8$ supergravity
[9] that was obtained by dimensional reduction of the unique 11D supergravity
[10]. The ultraviolet properties of these theories are still under
investigation to understand whether all supergravities must necessarily be
ultraviolet divergent, as known symmetry arguments seem to suggest, see [11]
and references therein. On the other hand, the various supergravity theories
appear as low energy limits of string theory [12, 13], whose finiteness is
related to its being a theory of extended objects rather than point-like
particles.

In this work, we focus on pure gravity with cosmological constant with the aim
to investigate further the structure of the diverging terms in the one-loop
effective action. Evaluated on-shell, these diverging terms are gauge-
invariant and characterize unambiguously the theory. For instance, they could
serve as a benchmark for verifying alternative approaches to perturbative
quantum gravity, and thus their precise expression should be known explicitly.
Our main contribution is the determination of the Seleey-DeWitt coefficient
$a_{3}(D)$ of perturbative quantum gravity that parametrizes a class of
divergences that start to appear in $D\geq 6$ dimensions. It has not been
reported in the literature in its full generality so far. On the other hand,
the coefficients $a_{n}(D)$ for $n=0,1,2$ are already known and have been
cross-checked with different methods [14, 15, 16].

In our endeavor to determine the coefficient $a_{3}(D)$, we employ two
distinct approaches: the worldline formalism and the heat kernel method. The
first one consists in using the $\mathcal{N}=4$ spinning particle, which
provides a first-quantized description of the graviton. It correctly describes
the graviton propagating on Einstein manifolds [17] and it was employed in
[14] to construct a worldline representation of the one-loop effective action
for quantum gravity. The worldline representation involves computing the path
integral of the $\mathcal{N}=4$ spinning particle on a circle. We perform the
perturbative expansion of this path integral and compute it up to the order
required to determine the coefficient $a_{3}(D)$. Then, we employ a second
method based on the time-honored heat kernel representation of the one-loop
effective action of quantum gravity [18, 19, 20]. In this approach, we
evaluate the heat kernel coefficients by taking on-shell the background
metric, i.e. inserting metrics corresponding to Einstein spaces. This second
method yields the same coefficient $a_{3}(D)$ obtained previously. The
agreement between the two methods provides a robust consistency check for both
the new coefficient and the worldline $\mathcal{N}=4$ representation.

The paper is organized as follows. In section 2, we provide a description of
the worldline formalism. Specifically, we focus on the path integral for the
$\mathcal{N}=4$ spinning particle and show how it can be used to compute the
on-shell counterterms in quantum gravity. We compute the explicit counterterms
related to the coefficients $a_{n}(D)$ for $n=0,1,2,3$ using worldline
perturbation theory. Section 3 employs the heat kernel method in one-loop
quantum gravity to derive the same counterterms, thus providing a cross-check
of our results. In section 4, we discuss the one-loop divergences for quantum
gravity in four and six dimensions, providing further consistency with the
existing literature. Finally, our conclusions are presented in section 5. The
appendices contain useful formulae for geometric quantities on Einstein spaces
and detailed explanations of the computational procedures employed.

##  2 Worldline formalism

In this section, we review the representation of the one-loop effective action
for pure Einstein-Hilbert gravity in the worldline formalism. The most elegant
way of obtaining such a representation is to consider a relativistic spinning
particle with four local supersymmetries on the worldline, check that its
spectrum coincides with that of the graviton in $D$ dimensions — especially
when the coupling to a background metric is introduced — and then path
integrate the model on the circle to obtain the desired one-loop effective
action for the graviton as a functional of the background metric. Along the
way, one finds that quantum consistency of the model requires the background
metric to satisfy Einstein’s equations of motion. This approach for describing
the graviton shares many analogies with string theory: they are both first
quantized models and both of them require the allowed background fields to be
on-shell. We shall not present here all the details leading to the
construction of the worldline representation of the effective action for pure
quantum gravity, for which we refer to [14], but we review the main ideas that
have led to that result.

Massless relativistic particles carrying spin $s$ have a description in terms
of mechanical models with ${\cal N}=2s$ local supersymmetries on the
worldline, as suggested in [21] and explicitly constructed in [22, 23]. These
models are described by the worldline particle coordinates $x^{\mu}$ together
with ${\cal N}$ real fermionic superpartners $\psi^{\mu}_{i}$ with
$i=1,\dots,{\cal N}$, introduced to describe the spin degrees of freedom of
the particle. Unitarity of the model requires the $\mu=0$ components of these
variables to be non-physical. The ${\cal N}$-extended local supersymmetry is
needed precisely to compensate for this redundancy in a relativistic invariant
manner. A further gauging of the R-symmetry group SO($\cal N$) that rotates
the $\cal N$ supercharges is optional as it is not required by unitarity. It
is generically used to constrain the model to have the minimal amount of
degrees of freedom and deliver pure spin $s$ states. This model was path
integrated on a circle in [24] to verify that it propagates correctly the
degrees of freedom of a relativistic particle of spin $s$. While everything
works fine in flat spacetimes, coupling to background fields, and in
particular to curved spacetimes, proved more difficult to achieve, which is
somehow expected for particles of sufficiently high spin. Early results were
obtained in [25, 26], where some obstructions were circumvented allowing for
specific couplings to (A)dS and conformally flat spaces. Then, a crucial
result was obtained in [27], where it was shown how to use BRST methods to
extend the case of spin 1 to include non-abelian couplings, thus providing a
first-quantized description of the gluon. Using similar BRST techniques, in
[17] it was found how the spin 2 massless particle could be coupled to
background metrics that satisfy Einstein’s equation. This guarantees that the
first-quantized graviton can propagate consistently on Einstein spacetimes,
where the Ricci tensor is proportional to the metric

$R_{\mu\nu}=\lambda g_{\mu\nu}$ (2.1)

with constant $\lambda$, thus admitting a cosmological constant of indefinite
sign. The BRST construction was crucial to develop a path integral
quantization delivering a worldline representation of the one-loop effective
action for quantum gravity [14]. Path integrals for one-dimensional nonlinear
sigma models, such as the one associated with the $\mathcal{N}=4$ spinning
particle in curved spaces, require counterterms that are related to the
regularization scheme used in defining the path integral itself [28, 29]. The
counterterm used in [14] was the one associated with wordline dimensional
regularization and was effectively valid only for $D=4$. Its extension to
arbitrary $D$ dimensions was constructed in [15] and it is the one that we use
here below.

Alternative worldline representations of the effective action for quantum
gravity are possible, see for instance [30, 16]. They are close in spirit to
the heat kernel approach employed in Section 4. For their formulation they
need direct inputs from the associated QFT, just as it happens in the heat
kernel method. In this sense, they are not independent of the second quantized
theory and will not be used in this section.

As anticipated, we refer to the aforementioned studies for details on the
analysis of the spinning particle action and quantization, proceeding now with
the construction of the worldline representation of the gravitational
effective action.

###  2.1 The worldloop path integral

The one-loop effective action $\Gamma[g_{\mu\nu}]$ for pure gravity
corresponds to the path integral of the $\mathcal{N}=4$ spinning particle
action $S[X,G;g_{\mu\nu}]$ on worldlines with the topology of the circle
$S^{1}$, also called the “worldloop”, and takes the schematic form

$\Gamma[g_{\mu\nu}]=\int_{S^{1}}\frac{\mathcal{D}G\,\mathcal{D}X}{\mathrm{Vol(Gauge)}}\,{\rm
e}^{-S[X,G;g_{\mu\nu}]}\ .$ (2.2)

The particle Euclidean action depends on the worldline gauge fields
$G=\left(e,\chi,\bar{\chi},a\right)$ and coordinates with supersymmetric
partners $X=\left(x,\psi,\bar{\psi}\right)$, while the overcounting from
summing over gauge equivalent configurations is formally taken into account by
dividing by the volume of the gauge group. The four real fermionic partners of
the coordinates have here been cast more conveniently into a pair of complex
fermionic variables $\psi$ and $\bar{\psi}$, as well as the worldline
gravitinos $\chi$ and $\bar{\chi}$. As implied by the BRST analysis, the gauge
symmetries are consistent only when the background metric $g_{\mu\nu}$ is on-
shell, i.e. satisfies eq. (2.1). Explicitly, the effective action (2.2) is
related to a partially gauge-fixed version of the $\mathcal{N}=4$ spinning
particle path integral $Z(T)$ through a Schwinger representation, and given by

$\Gamma[g_{\mu\nu}]=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T}Z(T)$ (2.3)

where $T$ is often called the Schwinger proper time, a modulus whose
integration arises from the gauge-fixing of the einbein $e$ on the circle. In
the present work, we only mention some of the important technicalities, namely
the gauging of a parabolic subgroup of the $SO(\mathcal{N})$ $R$-symmetry
group, the choice of the aforementioned gauge fixing of the worldline action,
and the regularization of the nonlinear supersymmetric sigma model, skipping
the details while referring to previous work for the reader interested in them
[14, 15]. We choose to focus our discussion on the perturbative computation of
the path integral providing only a few remarks when needed. The partition
function $Z(T)$, upon gauge fixing, takes the following explicit form

$\displaystyle Z(T)$
$\displaystyle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}\int_{0}^{2\pi}\frac{d\phi}{2\pi}\;P(\theta,\phi)\;\int_{{}_{\rm
PBC}}\hskip-11.38092pt{D}xDaDbDc\int_{{}_{\rm
ABC}}\hskip-11.38092ptD\bar{\psi}D\psi\,e^{-S[X;g_{\mu\nu}]}\ ,$ (2.4)

where $P(\theta,\phi)$ is the measure on the moduli space $(\theta,\phi)$
generated by the gauge fixing and which implements the correct projection on
the physical graviton Hilbert space in $D$ dimensions. The worldline variables
$X=\left(x,\psi,\bar{\psi},a,b,c\right)$ now include bosonic $a$ and fermionic
$(b,c)$ “metric ghosts”, introduced in order to keep translational invariance
of the path integral measure and which renormalize potentially divergent
worldline diagrams [28]. The path integral over bosonic variables and metric
ghosts is evaluated by fixing periodic boundary conditions (PBC), while the
fermionic path integral is performed by choosing antiperiodic boundary
conditions (ABC) on each flavor of fermionic fields $\psi^{a}_{i}$, with the
internal index $i$ taking values $i=1,2$.111Due to the complex combination of
the original $\mathcal{N}=4$ real fermions. The gauge-fixed nonlinear sigma
model action reads222The bosonic coordinates are understood to be shifted as
$\dot{x}^{\mu}\dot{x}^{\nu}\rightarrow\dot{x}^{\mu}\dot{x}^{\nu}+a^{\mu}a^{\nu}+b^{\mu}c^{\nu}$.
This shift implements into (2.5) the ghost action $S_{\rm{gh}}[x,a,b,c]=\int
d\tau\tfrac{1}{4T}g_{\mu\nu}(x)(a^{\mu}a^{\nu}+b^{\mu}c^{\nu})$ which allows
for the exponentiation of the determinant factor hidden inside the path
integral measure on a curved spacetime, i.e.
$\mathcal{D}x=\prod_{\tau}d^{D}x(\tau)\sqrt{g(x(\tau))}={D}x\int
DaDbDc\;e^{-S_{\rm{gh}}}\ ,$ where $Dx$, $Da$, $Db$, and $Dc$ are the standard
translational invariant measures. In particular, these metric ghosts create
worldline divergences that compensate for the divergences generated by
correlators of the $\dot{x}^{\mu}$’s. Divergences formally cancel out and one
is left with a finite theory, whose remaining ambiguities are taken care of by
choosing a regularization scheme with a corresponding counterterm that remains
finite.

$\displaystyle S[X;g_{\mu\nu}]=\int d\tau\Big{[}$
$\displaystyle\frac{1}{4T}g_{\mu\nu}(x)\,\dot{x}^{\mu}\dot{x}^{\nu}+\bar{\psi}^{ai}\left(\delta_{i}^{j}D_{\tau}-\hat{a}_{i}^{j}\right){\psi}_{aj}-TR_{abcd}(x)\,\bar{\psi}^{a}\cdot\psi^{b}\bar{\psi}^{c}\cdot\psi^{d}-T\,\mathcal{V}(x)\Big{]}\
,$ (2.5)

where we use flat indices on the worldline complex fermions $\psi^{a}_{i}$ and
denoted the covariant derivative with spin connection $\omega_{\mu ab}$ acting
on the fermions by
$D_{\tau}\psi^{a}_{i}=\partial_{\tau}\psi^{a}_{i}+\dot{x}^{\mu}\omega_{\mu}{}^{a}{}_{b}(x)\psi^{b}_{i}$.
We also used a dot to indicate contraction on the internal indices and denoted

$\hat{a}_{i}^{j}=\left(\begin{array}[]{cc}\theta&0\\\ 0&\phi\\\
\end{array}\right)$ (2.6)

the gauge-fixed values of the worldline gauge fields acting on the fermions
and related to the gauging of the parabolic subgroup of the $R$-symmetry
group. The angles $\theta$ and $\phi$ are precisely the two leftover moduli
remaining after the gauge-fixing procedure.333As discussed in [14], the
gauging of the parabolic subgroup allows the one-loop measure for the path
integral to be modified so that it projects _exactly_ onto the graviton state.
Alternatively, the gauging of the entire SO($4$) group would result in the
graviton plus unwanted contributions of topological nature, a case that might
be worth studying but is beyond the scope of our work.

A few comments are in order. The theory described by (2.5) should be seen as a
one-dimensional field theory living on the worldline with the bosonic fields
$x^{\mu}(\tau)$, the embedding of the worldline into spacetime, taking values
in a $D$-dimensional target space $\cal{M}$. On the worldline, one usually
finds it convenient to rescale the parameter $\tau$ to take values on the
finite interval $I\equiv[0,1]$. A crucial role is played by the scalar
potential term $\mathcal{V}$ of quantum origin [15]. It is necessary since it
contains the counterterm required by the regularization scheme one decides to
use to define the path integral and an additional potential needed to achieve
nilpotency of the BRST charge at the quantum level. The latter condition
requires a value of $V_{\rm BRST}=\frac{2}{D}R$ in the Hamiltonian constraint
[17]. Regarding the former, in the present work we adopt dimensional
regularization (DR) on the worldline, as used for instance in [31, 32, 33] in
similar contexts, while reading from [34] the counterterm $V_{\rm
CT}=-\frac{1}{4}R$ needed for the case of four supersymmetries, thus producing
an effective potential

$\mathcal{V}=V_{\rm BRST}+V_{\rm
CT}=\left(\frac{2}{D}-\frac{1}{4}\right)R\equiv\Omega R$ (2.7)

which indeed is the one used in [15]. Finally, for computational purposes, it
is convenient to rewrite the angular integrations over the moduli $\theta$ and
$\phi$ in the complex plane. This can be achieved by introducing the Wilson
variables $z\equiv e^{i\theta}$ and $\omega\equiv e^{i\phi}$, so to recast the
partition function as

$\displaystyle Z(T)$ $\displaystyle=\oint\frac{dz}{2\pi i}\frac{d\omega}{2\pi
i}\;P(z,\omega)\;\int_{{}_{\rm PBC}}\hskip-11.38092pt{D}xDaDbDc\int_{{}_{\rm
ABC}}\hskip-11.38092ptD\bar{\psi}D\psi\,e^{-S[X;g_{\mu\nu}]}\ ,$ (2.8)

where the modular integration is performed over the circle $|z|=1$, with the
singular point $z=-1$ pushed out of the contour,444Poles at $z=-1$ arise when
computing perturbative corrections and are excluded by this prescription. The
same goes for $\omega$. Discussion on the regulated contour of integration for
the modular parameters can be found in [35]. and with the measure on the
moduli space being now

$P(z,\omega)=\frac{1}{2}\frac{(z+1)^{D-2}}{z^{3}}\frac{(\omega+1)^{D-2}}{\omega^{3}}(z-\omega)^{2}(z\omega-1)\
.$ (2.9)

In the next section, we set up the perturbative expansion for small values of
$T$ of the path integral (2.8), such that the full $T^{n}$ correction is an
$(n+1)$-loop expansion in the worldline theory, which will allow us to
identify the divergences in the effective action of pure gravity.

###  2.2 Setting up the perturbative expansion

Having at hand a path integral representation for the effective action, it is
possible to set up the perturbative expansion around the free theory. However,
there are two issues to take care of in order to be able to perform
calculations. The first involves factorizing out the zero modes in the kinetic
operator in (2.5). Zero modes appear perturbatively once expanding around a
constant metric and when considering periodic boundary conditions. To this
task, we parametrize the bosonic coordinates of the circle (interpreted as a
parametrization of the particle paths in target space) as

$x^{\mu}(\tau)=x_{0}^{\mu}+q^{\mu}(\tau)\ ,$ (2.10)

thus describing all loops in spacetime with a fixed base point $x^{\mu}_{0}$
(the zero mode that is integrated over only at the end) plus quantum
fluctuations with vanishing Dirichlet boundary conditions (DBC), indicated by
$q^{\mu}(\tau)$ and thus satisfying $q^{\mu}(0)=q^{\mu}(1)=0$. Note that the
fermionic coordinates have no zero modes, due to their antiperiodic boundary
conditions. The second issue consists in expanding in Riemann normal
coordinates (RNC) centered around $x^{\mu}_{0}$, so to write the metric tensor
and the spin connection as follows [31, 36]

$\displaystyle\begin{split}g_{\mu\nu}(x(\tau))&=g_{\mu\nu}+\frac{1}{3}R_{\alpha\mu\nu\beta}q^{\alpha}q^{\beta}+\frac{1}{6}\nabla_{\gamma}R_{\alpha\mu\nu\beta}q^{\alpha}q^{\beta}q^{\gamma}+R_{\alpha\beta\mu\nu\gamma\delta}q^{\alpha}q^{\beta}q^{\gamma}q^{\delta}\\\
&\phantom{=}+\frac{1}{315}R_{\mu\alpha\beta}{}^{\sigma}R_{\sigma\gamma\delta}{}^{\lambda}R_{\lambda\tau\epsilon\nu}q^{\alpha}q^{\beta}q^{\gamma}q^{\delta}q^{\tau}q^{\epsilon}+\mathcal{O}(q^{6})\end{split}$
(2.11) $\displaystyle\begin{split}\omega_{\mu
ab}(x(\tau))&=\frac{1}{2}R_{\alpha\mu
ab}q^{\alpha}+\frac{1}{3}\nabla_{\alpha}R_{\beta\mu
ab}q^{\alpha}q^{\beta}+\frac{1}{8}\nabla_{\alpha}\nabla_{\beta}R_{\gamma\mu
ab}+\frac{1}{24}R^{\tau}{}_{\alpha\beta\mu}R_{\gamma\tau
ab}q^{\alpha}q^{\beta}q^{\gamma}+\mathcal{O}(q^{4})\end{split}$ (2.12)

where

$R_{\alpha\beta\mu\nu\gamma\delta}=\frac{1}{20}\nabla_{\delta}\nabla_{\gamma}R_{\alpha\mu\nu\beta}+\frac{2}{45}R_{\alpha\mu}{}^{\sigma}{}_{\beta}R_{\gamma\sigma\nu\delta}$
(2.13)

and where we only kept the terms needed to obtain a perturbative expansion to
order $T^{3}$. In (2.11)-(2.12) and henceforth, unless specified otherwise, we
intend all tensor structures to be evaluated at the initial point
$x^{\mu}_{0}$, thus factorizing out their dependence upon the worldline
bosonic variables $q^{\mu}(\tau)$. Finally, the Riemann tensor appearing in
the four-fermions interaction in (2.5) has to be Taylor expanded as well,
around the same reference point $x^{\mu}_{0}$. The perturbative expansion of
the path integral (2.8) reads

$Z(T)=\oint\frac{dz}{2\pi i}\frac{d\omega}{2\pi i}\;P(z,\omega)\;\int
d^{D}x_{0}\frac{\sqrt{g(x_{0})}}{\left(4\pi
T\right)^{\frac{D}{2}}}\Big{\langle}e^{-S_{\rm int}}\Big{\rangle}\ ,$ (2.14)

factorizing out for convenience the $\sqrt{g(x_{0})}$ arising from the free
ghost part of the action after functional integrating, together with
$\left(4\pi T\right)^{-\frac{D}{2}}$ arising from the free particle path
integral. The remaining expectation value is to be evaluated using the Wick
theorem on the free path integral, with the free action being given by the
quadratic part of (2.5), namely

$S_{0}[X]=\int
d\tau\left[\frac{1}{4T}g_{\mu\nu}\left(\dot{x}^{\mu}\dot{x}^{\nu}+a^{\mu}a^{\nu}+b^{\mu}c^{\nu}\right)+\bar{\psi}^{ai}\left(\delta_{i}^{j}\,\partial_{\tau}-\hat{a}_{i}\,^{j}\right)\psi_{aj}\right]\
,$ (2.15)

from which one obtains the worldline propagators of the theory, reported in
appendix B.1. Higher order terms form the interacting action $S_{\rm int}$, to
be analyzed later on. It is possible to recast (2.14) in a more compact form
introducing the double expectation value of the interacting action
$\langle\hskip-1.42271pt\langle\cdots\rangle\hskip-1.42271pt\rangle$, namely
the average over the path integral and over the moduli space parametrized by
$z$ and $w$

$\Big{\langle}\hskip-2.84544pt\Big{\langle}e^{-S_{\rm
int}}\Big{\rangle}\hskip-2.84544pt\Big{\rangle}=\oint\frac{dz}{2\pi
i}\frac{d\omega}{2\pi i}\;P(z,\omega)\Big{\langle}e^{-S_{\rm
int}}\Big{\rangle}\ .$ (2.16)

Identifying the expectation values defined above as

$\displaystyle\Big{\langle}e^{-S_{\rm
int}}\Big{\rangle}=\sum_{n=0}^{\infty}a_{n}(D,z,\omega)\,T^{n}\quad\longrightarrow\quad\Big{\langle}\hskip-2.84544pt\Big{\langle}e^{-S_{\rm
int}}\Big{\rangle}\hskip-2.84544pt\Big{\rangle}=\sum_{n=0}^{\infty}a_{n}(D)\,T^{n}\
,$ (2.17)

allows us to rearrange the path integral (2.14) so as to make explicit the
Seeley-DeWitt coefficients arising from the perturbative expansion

$\displaystyle Z(T)=\int\frac{d^{D}x_{0}}{\left(4\pi
T\right)^{\frac{D}{2}}}\sqrt{g(x_{0})}\left[a_{0}(D)+a_{1}(D)\,T+a_{2}(D)\,T^{2}+a_{3}(D)\,T^{3}+\mathcal{O}(T^{4})\right]\
.$ (2.18)

One can recognize that, while in the above sum $a_{0}(D,z,\omega)=1$, its
projected partner

$a_{0}(D)=\langle\hskip-1.42271pt\langle
1\rangle\hskip-1.42271pt\rangle=\frac{1}{2}\oint\frac{dz}{2\pi
i}\frac{d\omega}{2\pi
i}\frac{(z+1)^{D-2}}{z^{3}}\frac{(\omega+1)^{D-2}}{\omega^{3}}(z-\omega)^{2}(z\omega-1)=\frac{D(D-3)}{2}$
(2.19)

gives the massless graviton physical polarizations in $D$ spacetime dimensions
when using the correct measure $P(z,\omega)$ in eq. (2.9). We have now made
explicit our main task, namely to determine the Seeley-DeWitt coefficient
$a_{3}(D)$ in the perturbative expansion on Einstein spaces.

###  2.3 Outline of the computation

We will now give a brief outline of the procedure, delegating the details of
both the computation and regularization of potentially divergent diagrams to
appendix B. To systematically work out all perturbative contributions to the
desired order, one has first to identify and compute the connected worldline
diagrams arising from the path integral expansion. In order to do so, we
report the interacting action in (2.14) expanded to the desired order

$\displaystyle\begin{split}S_{\rm int}&=\int
d\tau\Bigg{[}\frac{1}{4T}\bigg{(}\frac{1}{3}R_{\alpha\mu\nu\beta}q^{\alpha}q^{\beta}+\frac{1}{6}\nabla_{\gamma}R_{\alpha\mu\nu\beta}q^{\alpha}q^{\beta}q^{\gamma}+\frac{1}{20}\nabla_{\delta}\nabla_{\gamma}R_{\alpha\mu\nu\beta}q^{\alpha}q^{\beta}q^{\gamma}q^{\delta}\\\
&+\frac{2}{45}R_{\alpha\mu}{}^{\sigma}\,_{\beta}R_{\gamma\sigma\nu\delta}q^{\alpha}q^{\beta}q^{\gamma}q^{\delta}+\frac{1}{315}R_{\mu\alpha\beta}{}^{\sigma}R_{\sigma\gamma\delta}{}^{\lambda}R_{\lambda\tau\epsilon\nu}q^{\alpha}q^{\beta}q^{\gamma}q^{\delta}q^{\tau}q^{\epsilon}\bigg{)}\left(\dot{q}^{\mu}\dot{q}^{\nu}+a^{\mu}a^{\nu}+b^{\mu}b^{\nu}\right)\\\
&+\left(\frac{1}{2}R_{\alpha\mu
ab}q^{\alpha}+\frac{1}{3}\nabla_{\alpha}R_{\beta\mu
ab}q^{\alpha}q^{\beta}+\frac{1}{8}\nabla_{\alpha}\nabla_{\beta}R_{\gamma\mu
ab}q^{\alpha}q^{\beta}q^{\gamma}+\frac{1}{24}R^{\tau}{}_{\alpha\beta\mu}R_{\gamma\tau
ab}q^{\alpha}q^{\beta}q^{\gamma}\right)\bar{\psi}^{a}\cdot\psi^{b}\,\dot{q}^{\mu}\\\
&-T\left(R_{abcd}+q^{\alpha}\nabla_{\alpha}R_{abcd}+\frac{1}{2}q^{\alpha}q^{\beta}\nabla_{\alpha}\nabla_{\beta}R_{abcd}\right)\bar{\psi}^{a}\cdot\psi^{b}\bar{\psi}^{c}\cdot\psi^{d}-T\,\Omega
R\Bigg{]}\ ,\end{split}$ (2.20)

which we write in a more compact form as

$\displaystyle\begin{split}S_{\rm int}=&\frac{1}{4T}\left(S_{\rm K1}+DS_{\rm
K1}+D^{2}S_{\rm K1}+S_{\rm K2}+S_{\rm K3}\right)\\\ &+S_{\rm C1}+DS_{\rm
C1}+D^{2}S_{\rm C1}+S_{\rm C2}\\\ &-T\left(S_{\rm F}+DS_{\rm F}+D^{2}S_{\rm
F}\right)+TS_{\rm V}\ ,\end{split}$ (2.21)

where the explicit expression of each term is obtained by comparing with the
previous expression, see also appendix B. Expanding the exponential in the
path integral to order $T^{3}$, we obtain the contributions that need to be
path-averaged and computed using Wick contractions. We list them here, leaving
their systematic analysis to appendix B.2, where we also show more details on
the intermediate steps of computation:

$\displaystyle\begin{split}e^{-S_{\rm int}}\Big{|}_{T^{3}}&=S_{\rm
KIN}-\frac{1}{8T}S_{\rm K1}S_{\rm C1}^{2}+S_{\rm C1}S_{\rm
C2}+\frac{1}{2}DS_{\rm C1}^{2}+D^{2}S_{\rm C1}S_{\rm C1}\\\
&\phantom{=}+\frac{1}{2}T^{2}S_{\rm C1}^{2}S_{\rm F}+\frac{1}{6}T^{3}S_{\rm
F}^{3}+\frac{1}{2}T^{2}DS_{\rm F}^{2}+T^{2}D^{2}S_{\rm F}S_{\rm F}\
,\end{split}$ (2.22)

where we have collectively denoted $S_{\rm KIN}$ the three-loop contributions
arising from the pure kinetic term, see (B.26). As an illustrative example, we
shall show how to compute a contribution containing all the main features of
this type of calculation, namely the correction arising from the spin-
connection vertex only555The subscript $01$ in (2.23) (as well as in appendix
B) serves as a shorthand notation to indicate henceforth
$\int_{01}\equiv\int_{0}^{1}d\tau\int_{0}^{1}d\sigma$ in worldline integrals.

$\displaystyle\langle S_{\rm C1}S_{\rm C2}\rangle=\frac{1}{48}R_{\alpha\mu
ab}R^{\tau}{}_{\beta\lambda\nu}R_{\rho\tau
cd}\int_{01}\langle\dot{q}^{\mu}_{0}q^{\alpha}_{0}\dot{q}^{\nu}_{1}q^{\beta}_{1}q^{\lambda}_{1}q^{\rho}_{1}\rangle\langle\bar{\psi}^{a}_{0}\cdot\psi_{0}^{b}\bar{\psi}^{c}_{1}\cdot\psi_{1}^{d}\rangle\
.$ (2.23)

The fermionic Wick contractions produce

$R_{\alpha\mu ab}R^{\tau}{}_{\beta\lambda\nu}R_{\rho\tau
cd}\,\langle\bar{\psi}^{a}_{0}\cdot\psi_{0}^{b}\bar{\psi}^{c}_{1}\cdot\psi_{1}^{d}\rangle=-\Delta_{\rm{AF}}{}_{ji}(\tau,\sigma)\Delta_{\rm{AF}}{}_{ij}(\sigma,\tau)R_{\alpha\mu
ab}R^{\tau}{}_{\beta\lambda\nu}R_{\rho\tau}{}^{ba}\ ,$ (2.24)

where we introduced the $\mathcal{N}=4$ fermionic propagator
$\Delta_{\rm{AF}\,}$. Evaluating then the bosonic contractions one gets
different Riemann tensor strings, which can be reduced using Bianchi identity,
namely

$\displaystyle
R^{\mu\nu\rho\sigma}R_{\mu\nu}{}^{\alpha\beta}R_{\rho\alpha\sigma\beta}=\frac{1}{2}R^{\mu\nu\rho\sigma}R_{\mu\nu}{}^{\alpha\beta}R_{\rho\sigma\alpha\beta}\;.$
(2.25)

Recall that all the calculations have to be further carried out with two
precautions, namely to use Einstein spaces simplifications (see appendix A)
and to perform the regularization using DR. Then, collecting all terms one
gets

$\left(\frac{1}{4}R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{R}{6D}R_{\mu\nu\rho\sigma}^{2}\right)\left(\leavevmode\hbox
to81.09pt{\vbox to41.26pt{\pgfpicture\makeatletter\hbox{\hskip
40.54518pt\lower-20.62825pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex[dot] (b) at
(0,0) {}; \vertex[dot](c)at(1.4,0){};
{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{10.99995pt}{8.91698pt}{19.91693pt}{19.91693pt}{19.91693pt}\pgfsys@curveto{30.91689pt}{19.91693pt}{39.83386pt}{10.99995pt}{39.83386pt}{0.0pt}\pgfsys@curveto{39.83386pt}{-10.99995pt}{30.91689pt}{-19.91693pt}{19.91693pt}{-19.91693pt}\pgfsys@curveto{8.91698pt}{-19.91693pt}{0.0pt}{-10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{}
{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-10.99995pt}{-8.91698pt}{-19.91693pt}{-19.91693pt}{-19.91693pt}\pgfsys@curveto{-30.91689pt}{-19.91693pt}{-39.83386pt}{-10.99995pt}{-39.83386pt}{0.0pt}\pgfsys@curveto{-39.83386pt}{10.99995pt}{-30.91689pt}{19.91693pt}{-19.91693pt}{19.91693pt}\pgfsys@curveto{-8.91698pt}{19.91693pt}{0.0pt}{10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\propag[fer] (b) to
[in=90, out=90,looseness=.5] (c); \propag[fer] (c) to [in=-90,
out=-90,looseness=.5] (b); \vertex[ringdot] (d) at (.2,.5) {};
\vertex[ringdot] (d) at (1.2,.5) {}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to81.09pt{\vbox to41.26pt{\pgfpicture\makeatletter\hbox{\hskip
40.54518pt\lower-20.62825pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex[dot] (b) at
(0,0) {}; \vertex[dot](c)at(1.4,0){};
{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{10.99995pt}{8.91698pt}{19.91693pt}{19.91693pt}{19.91693pt}\pgfsys@curveto{30.91689pt}{19.91693pt}{39.83386pt}{10.99995pt}{39.83386pt}{0.0pt}\pgfsys@curveto{39.83386pt}{-10.99995pt}{30.91689pt}{-19.91693pt}{19.91693pt}{-19.91693pt}\pgfsys@curveto{8.91698pt}{-19.91693pt}{0.0pt}{-10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{}
{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-10.99995pt}{-8.91698pt}{-19.91693pt}{-19.91693pt}{-19.91693pt}\pgfsys@curveto{-30.91689pt}{-19.91693pt}{-39.83386pt}{-10.99995pt}{-39.83386pt}{0.0pt}\pgfsys@curveto{-39.83386pt}{10.99995pt}{-30.91689pt}{19.91693pt}{-19.91693pt}{19.91693pt}\pgfsys@curveto{-8.91698pt}{19.91693pt}{0.0pt}{10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\propag[fer] (b) to
[in=90, out=90,looseness=.5] (c); \propag[fer] (c) to [in=-90,
out=-90,looseness=.5] (b); \vertex[ringdot] (d) at (.2,.5) {};
\vertex[ringdot] (d) at (1.2,-.5) {}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+z\rightarrow\omega\right)$
(2.26)

where in the graphical representation of the worldline Feynman diagrams full
dots denote vertices, an empty dot represents a derivative, a line denotes a
bosonic propagator, and oriented lines represent $z$-fermionic propagators.
The first diagram has to be regularized due to the singularity carried by the
bosonic propagator ${}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}$. In DR it
has to be $d$-dimensional extended as outlined in the following:

$\displaystyle\leavevmode\hbox to81.09pt{\vbox
to41.26pt{\pgfpicture\makeatletter\hbox{\hskip
40.54518pt\lower-20.62825pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex[dot] (b) at
(0,0) {}; \vertex[dot](c)at(1.4,0){};
{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{10.99995pt}{8.91698pt}{19.91693pt}{19.91693pt}{19.91693pt}\pgfsys@curveto{30.91689pt}{19.91693pt}{39.83386pt}{10.99995pt}{39.83386pt}{0.0pt}\pgfsys@curveto{39.83386pt}{-10.99995pt}{30.91689pt}{-19.91693pt}{19.91693pt}{-19.91693pt}\pgfsys@curveto{8.91698pt}{-19.91693pt}{0.0pt}{-10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{}
{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-10.99995pt}{-8.91698pt}{-19.91693pt}{-19.91693pt}{-19.91693pt}\pgfsys@curveto{-30.91689pt}{-19.91693pt}{-39.83386pt}{-10.99995pt}{-39.83386pt}{0.0pt}\pgfsys@curveto{-39.83386pt}{10.99995pt}{-30.91689pt}{19.91693pt}{-19.91693pt}{19.91693pt}\pgfsys@curveto{-8.91698pt}{19.91693pt}{0.0pt}{10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\propag[fer] (b) to
[in=90, out=90,looseness=.5] (c); \propag[fer] (c) to [in=-90,
out=-90,looseness=.5] (b); \vertex[ringdot] (d) at (.2,.5) {};
\vertex[ringdot] (d) at (1.2,.5) {}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\int d\tau d\sigma\
{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}(\tau,\sigma)\Delta(\tau,\sigma)\Delta(\tau,\tau)F(z,\tau,\sigma)F(z,\sigma,\tau)$
$\displaystyle\rightarrow\int
d^{d+1}t\,d^{d+1}s\,{}_{\alpha}\Delta_{\beta}(t,s)\,\Delta(s,t)\Delta(t,t)\mbox{Tr}\left(\gamma^{\alpha}F(z,t,s)\gamma^{\beta}F(z,s,t)\right)$
$\displaystyle=-$ $\displaystyle\int d^{d+1}t\,d^{d+1}s\
{}_{\alpha}\Delta(t,s)\,\Delta(t,s)_{\beta}\Delta(t,t)\mbox{Tr}\left(\gamma^{\alpha}F(z,t,s)\gamma^{\beta}F(z,s,t)\right)$
$\displaystyle-$ $\displaystyle\int d^{d+1}t\,d^{d+1}s\
{}_{\alpha}\Delta(t,s)\Delta(s,t)\Delta(t,t)\mbox{Tr}\left(F(z,t,s)\overset{\leftarrow}{\not{\partial}}F(z,s,t)+F(z,t,s)\overset{\rightarrow}{\not{\partial}}F(z,s,t)\right)$
$\displaystyle\rightarrow-$ $\displaystyle\int d\tau
d\sigma\,{}^{\bullet}\\!\Delta(\tau,\sigma)\Delta^{\hskip-1.42262pt\bullet}(\tau,\sigma)\Delta(\tau,\tau)F(z,\tau,\sigma)F(z,\sigma,\tau)$
$\displaystyle\hskip 8.5359pt=-\hskip 2.84544pt$
$\displaystyle\leavevmode\hbox to81.09pt{\vbox
to41.26pt{\pgfpicture\makeatletter\hbox{\hskip
40.54518pt\lower-20.62825pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex[dot] (b) at
(0,0) {}; \vertex[dot](c)at(1.4,0){};
{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{10.99995pt}{8.91698pt}{19.91693pt}{19.91693pt}{19.91693pt}\pgfsys@curveto{30.91689pt}{19.91693pt}{39.83386pt}{10.99995pt}{39.83386pt}{0.0pt}\pgfsys@curveto{39.83386pt}{-10.99995pt}{30.91689pt}{-19.91693pt}{19.91693pt}{-19.91693pt}\pgfsys@curveto{8.91698pt}{-19.91693pt}{0.0pt}{-10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{}
{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-10.99995pt}{-8.91698pt}{-19.91693pt}{-19.91693pt}{-19.91693pt}\pgfsys@curveto{-30.91689pt}{-19.91693pt}{-39.83386pt}{-10.99995pt}{-39.83386pt}{0.0pt}\pgfsys@curveto{-39.83386pt}{10.99995pt}{-30.91689pt}{19.91693pt}{-19.91693pt}{19.91693pt}\pgfsys@curveto{-8.91698pt}{19.91693pt}{0.0pt}{10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\propag[fer] (b) to
[in=90, out=90,looseness=.5] (c); \propag[fer] (c) to [in=-90,
out=-90,looseness=.5] (b); \vertex[ringdot] (d) at (.2,.5) {};
\vertex[ringdot] (d) at (1.2,-.5) {}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{z}{60(z+1)^{2}}$
(2.27)

where in reaching the last-but-one line we used the regulated Green equation
for the component $F$ of the fermionic propagator $\Delta_{\rm{AF}\,}$, see
eqs. (B.7), (B.16) and (B.17) of appendix B, and then removed the
regularization by sending the extra dimensions to zero ($d\to 0$). Finally,
taking into account the $\omega$ partner of the integrals in (2.27), one gets
the following result:

$\displaystyle\langle S_{\rm C1}S_{\rm
C2}\rangle=\left(\frac{1}{120}R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{R}{180D}R_{\mu\nu\rho\sigma}^{2}\right)\left(\frac{z}{(z+1)^{2}}+\frac{\omega}{(\omega+1)^{2}}\right)\;.$
(2.28)

Once having worked out systematically all the corrections up to and including
order $T^{3}$ related to the connected graphs, it only remains to exponentiate
them and, finally, Taylor expand in $T$ so as to reach the desired order and
get the full result.

###  2.4 Seeley-DeWitt coefficients

The calculation of the various terms in the perturbative expansion delivers
the following coefficients in the perturbative series (2.18), including the
newly found $a_{3}(D)$

$\displaystyle\begin{split}a_{0}(D)&=\frac{D(D-3)}{2}\end{split}$ (2.29)
$\displaystyle\begin{split}a_{1}(D)&=\frac{D^{2}-3D-36}{12}\;R\end{split}$
(2.30)
$\displaystyle\begin{split}a_{2}(D)&=\frac{5D^{3}-17D^{2}-354D-720}{720D}\;R^{2}+\frac{D^{2}-33D+540}{360}\;R_{\mu\nu\rho\sigma}^{2}\end{split}$
(2.31)
$\displaystyle\begin{split}a_{3}(D)&=\frac{35D^{4}-147D^{3}-3670D^{2}-13560D-30240}{90720D^{2}}\;R^{3}+\frac{7D^{3}-230D^{2}+3357D+12600}{15120D}\;R\,R_{\mu\nu\rho\sigma}^{2}\\\\[3.00003pt]
&\phantom{=}+\frac{17D^{2}-555D-15120}{90720}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{D^{2}-39D-1080}{3240}\;R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\
.\end{split}$ (2.32)

The above expressions are understood to be gauge-invariant, as they have been
calculated specifically on Einstein spaces for the reasons previously
discussed. As we shall discuss later, the newly computed coefficient
$a_{3}(D)$ plays a key role in quantum gravity in six dimensions, as it is
related to the logarithmic divergences, hence it will be useful to read its
value explicitly in this case as well. In $D=6$ it reduces to

$\displaystyle\left.a_{3}\right|_{D=6}$
$\displaystyle=-\frac{799}{11340}\;R^{3}+\frac{481}{1680}\;R\,R_{\mu\nu\rho\sigma}^{2}-\frac{991}{5040}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}-\frac{71}{180}\,R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\
.$ (2.33)

Also, its expression on maximally symmetric spaces (MSS) may be useful for
future reference. As we are not aware of any such calculation carried out in
the literature we list it here. Using the relations in appendix A, we find
that (2.32) on MSS collapses into

$\displaystyle a_{3}^{\rm
MSS}(D)=\frac{35D^{6}-217D^{5}-3257D^{4}-9239D^{3}+37470D^{2}+183672D-302400}{90720(D-1)^{2}D^{2}}\;R^{3}\
,$ (2.34)

yielding in $D=6$

$\displaystyle\left.a_{3}^{\rm MSS}\right|_{D=6}=-\frac{3181}{63000}\;R^{3}\
.$ (2.35)

####  2.4.1 Ghost and Graviton separately

For future comparison with the results of the heat kernel method calculation,
it is useful to project on the degrees of freedom of the ghost and graviton
respectively. To this task, we must use the right measures in the path
integral (2.14). These can be found generalizing to our case the worldline
partition functions of [30] leading us to

$\displaystyle P_{\rm gh}(z,\omega)=\frac{(z+1)^{D}}{z^{2}}\frac{1}{\omega}\
,$ (2.36) $\displaystyle P_{\rm gr}(z,\omega)=\frac{2(z+1)^{D}}{\omega
z^{2}}+\frac{(\omega+1)^{D-2}(z+1)^{D-2}(z-\omega)^{2}(\omega
z-1)}{2\omega^{3}z^{3}}\ .$ (2.37)

Indeed, note that

$P_{\rm gr}(z,\omega)-2\,P_{\rm gh}(z,\omega)=P(z,\omega)\ .$ (2.38)

It is therefore possible to obtain the contributions to the Seeley-DeWitt
coefficients coming only from the ghost (graviton) projecting onto the desired
Hilbert space via $P_{\rm gh}\,(P_{\rm gr})$. Let us emphasize that these
results are not new _per se_ since the heat kernel procedure requires
calculating them individually and then putting them together, as we shall see.
What is new here is the possibility of obtaining them also from the worldline
viewpoint of the ${\cal N}=4$ particle. Therefore, regarding the ghost we have

$\displaystyle\begin{split}a_{0}^{\rm gh}(D)&=D\end{split}$ (2.39)
$\displaystyle\begin{split}a_{1}^{\rm gh}(D)&=\frac{D+6}{6}\;R\end{split}$
(2.40) $\displaystyle\begin{split}a_{2}^{\rm
gh}(D)&=\frac{5D^{2}+58D+180}{360D}\;R^{2}+\frac{D-15}{180}\;R_{\mu\nu\rho\sigma}^{2}\end{split}$
(2.41) $\displaystyle\begin{split}a_{3}^{\rm
gh}(D)&=\frac{35D^{3}+588D^{2}+3512D+7560}{45360D^{2}}\;R^{3}+\frac{7D^{2}-62D-714}{7560D}\;R\,R_{\mu\nu\alpha\beta}^{2}\\\\[3.00003pt]
&\phantom{=}+\frac{17D-252}{45360}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{D-18}{1620}\;R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\
,\end{split}$ (2.42)

while concerning the graviton

$\displaystyle\begin{split}a_{0}^{\rm gr}(D)&=\frac{1}{2}D(D+1)\end{split}$
(2.43) $\displaystyle\begin{split}a_{1}^{\rm
gr}(D)&=\frac{1}{12}\left(D^{2}+D-12\right)\;R\end{split}$ (2.44)
$\displaystyle\begin{split}a_{2}^{\rm
gr}(D)&=\frac{\left(5D^{2}+3D-122\right)}{720}\;R^{2}+\frac{\left(D^{2}-29D+480\right)}{360}\;R_{\mu\nu\rho\sigma}^{2}\end{split}$
(2.45) $\displaystyle\begin{split}a_{3}^{\rm
gr}(D)&=\frac{35D^{3}-7D^{2}-1318D+488}{90720D}\;R^{3}+\frac{7D^{3}-202D^{2}+3109D+9744}{15120D}\;R\,R_{\mu\nu\alpha\beta}^{2}\\\\[3.00003pt]
&\phantom{=}+\frac{17D^{2}-487D-16128}{90720}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{D^{2}-35D-1152}{3240}\;R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\
.\end{split}$ (2.46)

One can easily see that the coefficient $a_{0}(D)$ correctly reproduces the
expected degrees of freedom in $D$ dimensions. Moreover, it is immediate to
check that by summing up the contributions as prescribed by (2.38) we obtain
the correct total coefficients (2.29)–(2.32).

##  3 Heat kernel method

The gauge-invariant coefficients computed from the $\mathcal{N}=4$ spinning
particle can be obtained in an equivalent but completely independent manner by
exploiting the heat kernel method. This is a well-known technique from
mathematical physics, which allows studying perturbatively second-order
differential operators on Riemannian manifolds [37], and it has been applied
extensively to quantum gravity starting from the work of DeWitt, [18, 19, 20],
as for example in [38, 39, 40, 41]. In this section, we briefly review the
ideas behind this technique and apply it to the case of Euclidean perturbative
quantum gravity, to reproduce the coefficients (2.30)–(2.32) and provide a
strong consistency check for our computations.

###  3.1 Heat kernel and one-loop effective action

It is well known that the one-loop effective action for a bosonic or fermionic
field theory living in a $D$-dimensional Euclidean space assumes the form (see
for example [40])

$\Gamma_{(1)}[\Phi]=\frac{1}{2}\log{\text{sDet}{\Delta}}=\frac{1}{2}{\text{sTr}\log{\Delta}}\
,$ (3.1)

where $\Delta$ is an elliptic second-order differential operator of the form

$\Delta=-\nabla_{\scriptscriptstyle\\!\\!({\cal A})}^{2}-V$ (3.2)

while “sDet” is the Berezin functional superdeterminant and “sTr” the
functional supertrace. The operator $\Delta$ is here taken to act on a scalar
field $\phi$ which carries a representation of the (non-abelian) gauge field
${\cal A}_{\mu}$, contained in the connection that defines the full covariant
derivative $\nabla^{\scriptscriptstyle({\cal A})}_{\mu}=\nabla_{\mu}+{\cal
A}_{\mu}$. It has an associated gauge field strength $\Omega_{\mu\nu}$,
defined by the commutator of the covariant derivatives on the scalar field
$\phi$, $[\nabla^{\scriptscriptstyle({\cal
A})}_{\mu},\nabla^{\scriptscriptstyle({\cal
A})}_{\nu}]\phi=\Omega_{\mu\nu}\phi$. The Laplacian is defined as usual by
$\nabla_{\scriptscriptstyle\\!\\!({\cal
A})}^{2}=g^{\mu\nu}\nabla^{\scriptscriptstyle({\cal
A})}_{\mu}\nabla^{\scriptscriptstyle({\cal A})}_{\nu}$ and we consider a
potential $V$ which is matrix valued, just like the gauge field ${\cal
A}_{\mu}$. Thus, the elliptic second-order differential operator $\Delta$
depends on the metric $g_{\mu\nu}$ and on the matrix valued potential ${\cal
A}_{\mu}$ and $V$.

Using Schwinger proper-time parametrization, we can rewrite (3.1) as

$\Gamma_{(1)}=\frac{1}{2}{\text{sTr}\log{\Delta}}=-\frac{1}{2}\int_{0}^{\infty}\frac{\mbox{d}T}{T}\
\text{sTr}\exp{(-T\Delta)}\ ,$ (3.3)

where an infinite additive constant is neglected, as usual. The operator
$\exp{(-T\Delta)}$ is known as the heat semigroup of the operator $\Delta$,
and its integral kernel is the heat kernel $U(T;x,x^{\prime})$ (corresponding
to the matrix elements $\langle x|\exp{(-T\Delta)}|x^{\prime}\rangle$ in
quantum mechanical terms), which in the following will always be considered at
coinciding points $x^{\prime}\to x$. Plugging the explicit form of $\Delta$
given by (3.2) inside (3.3), and introducing a mass term by shifting
$\Delta\to\Delta+m^{2}$, so to have an infrared regulating mass if the
original field was massless, we have

$\Gamma_{(1)}=-\frac{1}{2}\int_{0}^{\infty}\frac{\mbox{d}T}{T}\
\exp{(-Tm^{2})}\int\mbox{d}^{D}x\ \sqrt{g}\ \text{str}\,{U(T;x,x)}\ .$ (3.4)

Note that the leftover supertrace “str” is now to be performed only over the
remaining discrete indices carried by the representation of the field $\phi$
on which $\Delta$ acts upon.

At this point, we employ the heat kernel expansion for small Euclidean time
$T\to 0$, at coinciding points, which takes the form [18, 19, 20]

$U(T;x,x)\sim(4\pi T)^{-\frac{D}{2}}\sum_{j=0}^{\infty}T^{j}a_{j}(x)\ ,$ (3.5)

where the heat kernel coefficients $a_{j}(x)$, also known as Seeley-DeWitt
coefficients, can be expressed in terms of the metric and gauge invariants of
the manifold. With the aid of (3.5), we are able to identify the divergences
of the one-loop effective action (3.4) precisely with a subset of the Seeley-
DeWitt coefficients, namely the ones that produce a divergence in the proper
time integration at $T\to 0$ in

$\Gamma_{(1)}=-\frac{1}{2}\int_{0}^{\infty}\frac{\mbox{d}T}{T}\
\exp{(-Tm^{2})}\int\frac{\mbox{d}^{D}x\,\sqrt{g}}{(4\pi T)^{\frac{D}{2}}}\
\mbox{sTr}\,\sum_{j=0}^{\infty}T^{j}a_{j}(x)\ .$ (3.6)

For example, at $D=4$ the diverging terms are associated with the coefficients
$a_{0}(x),a_{1}(x),a_{2}(x)$, while at $D=6$ also $a_{3}(x)$ leads to an
additional divergence (the logarithmic divergence in that dimension). The
problem of finding the UV divergences of the effective action is then reduced
to the computation of Seeley-DeWitt coefficients for a generic theory, which
has already been carried out up to $a_{5}(x)$ [37]. For application to
perturbative quantum gravity, we will consider here the first four
coefficients, i.e. from $a_{0}(x)$ to $a_{3}(x)$. The fourth coefficient
$a_{3}(x)$, in particular, has been computed for the first time by Gilkey
[42], and later confirmed by Avramidi through a fully covariant method
[43].666It is important to note that the notation we employ here for the
Riemann tensor, Ricci tensor, and scalar is the same of [15, 44], see equation
(A.1), while [42, 41] adopt the opposite sign in the Riemann tensor and in its
contractions, $R_{\mu\nu}\equiv R_{\alpha\mu\nu}^{\ \ \ \ \alpha}$, so the
heat kernel coefficients (3.10)–(3.13) are to be modified accordingly.

We now list the general results for the coefficients corresponding to the
operator (3.2) up to $a_{3}(x)$, as taken from [42, 44]. For the sake of
brevity, it is useful to define as done in [44]

$a_{j}(x)\equiv\alpha_{j}(x)+\beta_{j}(x)\ ,$ (3.7)

where the first term $\alpha_{j}(x)$ comes from considering an exponentiated
form of the heat kernel series,

$\mbox{sTr}\,\left[\sum_{j=0}^{\infty}T^{j}a_{j}(x)\right]\equiv\mbox{sTr}\,\left[\exp{\left(\sum_{j=1}^{\infty}T^{j}\alpha_{j}(x)\right)}\right]\
,$ (3.8)

while the second one $\beta_{j}(x)$ is the remainder, which, up to the third
order, is evaluated as

$\beta_{0}=\beta_{1}=0\ ,\quad\beta_{2}=\frac{1}{2}\alpha_{1}^{2}\
,\quad\beta_{3}=\frac{1}{6}\alpha_{1}^{3}+\alpha_{1}\alpha_{2}\ .$ (3.9)

The coefficients $\alpha_{j}(x)$ are given by

$\displaystyle\begin{split}\alpha_{0}(x)&=\mathbbm{1}\end{split}$ (3.10)
$\displaystyle\begin{split}\alpha_{1}(x)&=\frac{1}{6}R\mathbbm{1}+V\end{split}$
(3.11)
$\displaystyle\begin{split}\alpha_{2}(x)&=\frac{1}{6}\nabla^{2}\left(\frac{1}{5}R\mathbbm{1}+V\right)+\frac{1}{180}\left(R_{\mu\nu\rho\sigma}^{2}-R_{\mu\nu}^{2}\right)\mathbbm{1}+\frac{1}{12}\Omega_{\mu\nu}^{2}\end{split}$
(3.12)
$\displaystyle\begin{split}\alpha_{3}(x)&=\frac{1}{7!}\left[18\nabla^{4}R+17(\nabla_{\mu}R)^{2}-2(\nabla_{\mu}R_{\nu\sigma})^{2}-4\nabla_{\mu}R_{\nu\sigma}\nabla^{\nu}R^{\mu\sigma}+9(\nabla_{\alpha}R_{\mu\nu\rho\sigma})^{2}-8R_{\mu\nu}\nabla^{2}R^{\mu\nu}\right.\\\
&\phantom{=}+12R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}R+12R_{\mu\nu\rho\sigma}\nabla^{2}R^{\mu\nu\rho\sigma}+\frac{8}{9}R_{\mu}{}^{\nu}R_{\nu}{}^{\sigma}R_{\sigma}{}^{\mu}-\frac{8}{3}R_{\mu\nu}R_{\rho\sigma}R^{\mu\rho\nu\sigma}\\\
&\phantom{=}\left.-\frac{16}{3}R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\rho\sigma\tau}+\frac{44}{9}R_{\mu\nu}{}^{\rho\sigma}R_{\rho\sigma}{}^{\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{80}{9}R_{\mu\nu\rho\sigma}R^{\mu\alpha\rho\beta}R^{\nu}{}_{\alpha}{}^{\sigma}{}_{\beta}\right]\mathbbm{1}\\\
&\phantom{=}+\frac{2}{6!}\left[8(\nabla_{\mu}\Omega_{\nu\sigma})^{2}+2(\nabla^{\mu}\Omega_{\mu\nu})^{2}+12\Omega_{\mu\nu}\nabla^{2}\Omega^{\mu\nu}-12\Omega_{\mu}{}^{\nu}\Omega_{\nu}{}^{\sigma}\Omega_{\sigma}{}^{\mu}+6R_{\mu\nu\rho\sigma}\Omega^{\mu\nu}\Omega^{\rho\sigma}-4R_{\mu\nu}\Omega^{\mu\sigma}\Omega^{\nu}{}_{\sigma}\right.\\\
&\phantom{=}\left.+6\nabla^{4}V+30(\nabla_{\mu}V)^{2}+4R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}V+12\nabla_{\mu}R\nabla^{\mu}V\right]\
.\end{split}$ (3.13)

They will be used in the next section.

###  3.2 Euclidean quantum gravity

Consider a $D$-dimensional Riemannian manifold $(\mathcal{M},\bm{G})$ equipped
with a metric tensor $\bm{G}$ having Euclidean signature. The starting point
for our treatment of gravity is the Einstein-Hilbert action,

$S_{\rm EH}[\bm{G}]=-\frac{1}{k^{2}}\int\mbox{d}^{D}x\
\sqrt{G}\left[R(\bm{G})-\Lambda\right]\ ,$ (3.14)

where $k^{2}\equiv 16\pi G_{\rm N}$, being $G_{\rm N}$ the Newton constant,
$R(\bm{G})$ is the Ricci scalar computed from $\bm{G}$, and
$G\equiv\left|\det{G_{\mu\nu}}\right|$. A cosmological constant $\Lambda$ has
also been included. By employing the background field method, we split the
metric tensor $\bm{G}$ into a fixed classical background $\bm{g}$ and “small”
quantum perturbations $\bm{h}$, namely:

$G_{\mu\nu}(x)=g_{\mu\nu}(x)+h_{\mu\nu}(x)\ .$ (3.15)

As a consequence of this splitting, the action (3.14) can be expanded in power
series in the fluctuations $\bm{h}$. Since we are interested in the one-loop
level of accuracy, we will be concerned with the second-order term in
$\bm{h}$, which reads

$\displaystyle S_{2}$
$\displaystyle=\int\mbox{d}^{D}x\sqrt{g}\left[-\frac{1}{4}h^{\mu\nu}\left(\nabla^{2}+2\Lambda-R\right)h_{\mu\nu}+\frac{1}{8}h\left(\nabla^{2}+2\Lambda-R\right)-\frac{1}{2}\left(\nabla^{\nu}h_{\mu\nu}-\frac{1}{2}\nabla_{\mu}h\right)^{2}\right.$
$\displaystyle\phantom{=\int\mbox{d}^{D}x\sqrt{g}+}\left.-\frac{1}{2}\left(h^{\mu\lambda}h_{\lambda}{}^{\nu}-hh^{\mu\nu}\right)R_{\mu\nu}-\frac{1}{2}h^{\mu\lambda}h^{\nu\rho}R_{\mu\nu\lambda\rho}\right]\
.$ (3.16)

It is important to note that in (3.2) the Ricci tensor
$R_{\mu\nu}=R_{\mu\nu}(\bm{g})$ and scalar $R=R(\bm{g})$, as well as covariant
derivatives $\nabla_{\mu}=\nabla_{\mu}(\bm{g})$, are computed with respect to
the background metric $\bm{g}$. The gauge symmetries acting on $\bm{h}$ and
leaving the background metric $\bm{g}$ invariant are then BRST quantized by
introducing the ghost $c$ and antighost $\overline{c}$ fields, and adding to
the action the Slavnov variation of de Donder gauge-fixing function
$f_{\mu}=\nabla^{\nu}h_{\mu\nu}-\frac{1}{2}\nabla_{\mu}h$, see for example
[30] for further details. The final result is

$S_{2}[\bm{h},c,\overline{c}]=S_{\rm gr}[\bm{h}]+S_{\rm gh}[c,\overline{c}]\
,$ (3.17)

where

$\displaystyle\begin{split}S_{\rm
gr}[\bm{h}]&=\int\mbox{d}^{D}x\sqrt{g}\left[-\frac{1}{4}h^{\mu\nu}\left(\nabla^{2}+2\Lambda-R\right)h_{\mu\nu}+\frac{1}{8}h\left(\nabla^{2}+2\Lambda-R\right)\right.\\\
&\phantom{=\int\mbox{d}^{D}x\sqrt{g}+}\left.-\frac{1}{2}\left(h^{\mu\lambda}h_{\lambda}{}^{\nu}-hh^{\mu\nu}\right)R_{\mu\nu}-\frac{1}{2}h^{\mu\lambda}h^{\nu\rho}R_{\mu\nu\lambda\rho}\right]\
,\end{split}$ (3.18) $\displaystyle\begin{split}S_{\rm
gh}[c,\overline{c}]&=\int\mbox{d}^{D}x\sqrt{g}\
\bar{c}^{\mu}\left(\nabla^{2}c_{\mu}+R_{\mu\nu}c^{\nu}\right)\ .\end{split}$
(3.19)

We are now able to identify from the actions $S_{\rm gr}$ and $S_{\rm gh}$ the
invertible kinetic operators for the graviton and ghost fields, denoted by
$F_{\mu\nu\alpha\beta}$ and $\mathcal{F}_{\mu\nu}$, respectively. By setting

$S_{\rm gr}[\bm{h}]=\int\mbox{d}^{D}x\ \sqrt{g}\
\frac{1}{2}h_{\mu\nu}F^{\mu\nu\alpha\beta}h_{\alpha\beta}\qquad\mbox{and}\qquad
S_{\rm gh}[c,\overline{c}]=\int\mbox{d}^{D}x\ \sqrt{g}\
\bar{c}_{\mu}\mathcal{F}^{\mu}{}_{\nu}c^{\nu}\ ,$ (3.20)

and exploiting the properties of Einstein spaces, we find

$\displaystyle F_{\mu\nu}{}^{\alpha\beta}$
$\displaystyle=-\frac{1}{2}\left(\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}+\delta_{\mu}^{\beta}\delta_{\nu}^{\alpha}\right)\nabla^{2}-R_{\mu}{}^{\alpha}{}_{\nu}{}^{\beta}-R_{\mu}{}^{\beta}{}_{\nu}{}^{\alpha}$
(3.21) $\displaystyle\mathcal{F}^{\mu}_{\ \nu}$
$\displaystyle=\delta^{\mu}_{\nu}\left(\nabla^{2}+\frac{1}{D}R\right)\ ,$
(3.22)

where the graviton operator indices are raised and lowered with the DeWitt
supermetric

$\gamma^{\mu\nu\alpha\beta}\equiv\frac{1}{4}\left(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}-g^{\mu\nu}g^{\alpha\beta}\right)\
,\quad\gamma_{\mu\nu\alpha\beta}=g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha}-\frac{2}{D-2}g_{\mu\nu}g_{\alpha\beta}\
.$ (3.23)

The reasons for immediately reducing to Einstein spaces — that is, to compute
the coefficients directly on-shell — are twofold. Firstly, this allows us a
direct comparison with the worldline results, even separately for ghost and
graviton, since the on-shell condition is forced by the quantum consistency of
the $\mathcal{N}=4$ spinning particle. Secondly, proceeding otherwise the
results would not be gauge-invariant, but rather would depend on the gauge
chosen, as expected for gauge theories, see for instance the recent analysis
carried out in [45].

Since (3.21)–(3.22) are elliptic second-order differential operators, they can
be treated within the heat kernel expansion, and the coefficients
(3.10)–(3.13) can be computed by identifying the explicit formulae for
$\mathbbm{1}$, $V$ and $\Omega_{\mu\nu}$. The ghost field configuration space
is $D$-dimensional, and by comparing (3.22) with (3.2), as well as recalling
that

$[\nabla_{\mu},\nabla_{\nu}]c^{\rho}=R_{\mu\nu}{}^{\rho}{}_{\sigma}c^{\sigma}\
,$ (3.24)

we conclude that the substitutions to be performed in the heat kernel
coefficients (3.10)–(3.13) are

$\begin{cases}\mathbbm{1}\ \leftrightarrow\ \delta^{\mu}_{\nu}\\\\[3.00003pt]
V\ \leftrightarrow\ \dfrac{1}{D}R\delta^{\mu}_{\nu}\\\\[5.0pt]
(\Omega_{\mu\nu})^{\rho}_{\ \sigma}\ \leftrightarrow\
R_{\mu\nu}{}^{\rho}{}_{\sigma}\ .\end{cases}$ (3.25)

Note that in this expression the indices $\mu$, $\nu$ label the different
elements of the gauge field strength $\Omega_{\mu\nu}$, which are $D\times D$
matrices whose components are given by the (spacetime) indices $\rho$,
$\sigma$. On the other hand, for the graviton field the configuration space is
$\frac{1}{2}D(D+1)$ dimensional (space of symmetric tensors) and the
substitutions to be performed are

$\begin{cases}\mathbbm{1}\ \leftrightarrow\ \delta_{\mu\nu}^{\ \ \
\alpha\beta}\\\ V\ \leftrightarrow\ \mathcal{V}_{\mu\nu}{}^{\alpha\beta}\\\
(\Omega_{\mu\nu})_{\rho\sigma}{}^{\alpha\beta}\ \leftrightarrow\
R_{\rho\sigma}{}^{\alpha\beta}{}_{\mu\nu}\end{cases}$ (3.26)

where

$\displaystyle\delta_{\mu\nu}{}^{\alpha\beta}$
$\displaystyle\equiv\frac{1}{2}\left(\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}+\delta_{\mu}^{\beta}\delta_{\nu}^{\alpha}\right)$
(3.27) $\displaystyle\mathcal{V}_{\mu\nu}{}^{\alpha\beta}$
$\displaystyle\equiv
R_{\mu}{}^{\alpha}{}_{\nu}{}^{\beta}+R_{\mu}{}^{\beta}{}_{\nu}{}^{\alpha}\ ,$
(3.28)

and the commutator is given by a symmetrized version of the Riemann tensor

$[\nabla_{\mu},\nabla_{\nu}]h_{\rho\sigma}=R_{\rho\sigma}{}^{\alpha\beta}{}_{\mu\nu}\,h_{\alpha\beta}\
,$ (3.29)

where

$R_{\rho\sigma}{}^{\alpha\beta}{}_{\mu\nu}\equiv\frac{1}{2}\left(\delta_{\rho}^{\alpha}R_{\sigma}{}^{\beta}{}_{\mu\nu}+\delta_{\rho}^{\beta}R_{\sigma}{}^{\alpha}{}_{\mu\nu}+\delta_{\sigma}^{\alpha}R_{\rho}{}^{\beta}{}_{\mu\nu}+\delta_{\sigma}^{\beta}R_{\rho}{}^{\alpha}{}_{\mu\nu}\right)\
.$ (3.30)

At this point, the computations are tedious but straightforward, see appendix
C for details. The final results for the ghost and graviton fields separately
are precisely the coefficients (2.39)–(2.42) and (2.43)–(2.46) obtained from
the worldline formalism. The total coefficients for the physical graviton,
according to the supertrace appearing in (3.6), as recognized also from
(2.38), are given by

$\mbox{Tr}\left[a_{j}\right]=\mbox{Tr}\left[a_{j}^{\rm
gr}\right]-2\mbox{Tr}\left[a_{j}^{\rm gh}\right]\ .$ (3.31)

Again, the results obtained reproduce the ones coming from worldline
computations (2.30)–(2.32), providing a strong cross-check for the correctness
of both.

##  4 On one-loop divergences of quantum gravity

The coefficients (2.30)–(2.32), which include the newly computed coefficient
$a_{3}(D)$, allow for further investigations of the issue of divergences in
the quantum theory of gravity. Thus, let us review some crucial results from
the literature and discuss how our newly-calculated coefficient $a_{3}(D)$
fits into the picture, thus providing us with additional confirmation of the
validity of our result. We focus our discussion on the spacetime of dimensions
$D=4$ and $D=6$, as in these cases there are no additional divergences on top
of the one we have already computed (new divergences start to appear from
$D=8$ onwards). The type of divergences arising in quantum gravity emerge
naturally from the representation of the one-loop effective action with a
short proper time expansion, which we can read both from the worldline
viewpoint (2.18) and from the heat kernel one (3.6):

$\Gamma[g_{\mu\nu}]=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T^{1+\frac{D}{2}}}\int\frac{d^{D}x_{0}}{\left(4\pi\right)^{\frac{D}{2}}}\sqrt{g(x_{0})}\left[a_{0}+a_{1}T+a_{2}T^{2}+a_{3}T^{3}+\mathcal{O}(T^{4})\right]\
.$ (4.1)

We are interested in studying the UV divergences that arise from the
$T\rightarrow 0$ limit of the proper time integration. Setting $D=4$ we
recognize that possible divergences arise from the coefficients
$a_{0},a_{1},a_{2}$, with $a_{2}$ being associated with the logarithmic
divergence. In $D=6$, also $a_{3}$ gives rise to an additional divergence, the
logarithmic one in that dimension.

One may wonder how to relate the $\frac{1}{\epsilon}$ pole of dimensional
regularization in QFT, widely present in the literature, with our situation.
To address this point, it is useful to evaluate the proper time integral term
by term in (4.1), to display the gamma function dependence. We find

$\int_{0}^{\infty}\frac{dT}{T^{1+\frac{D}{2}}}T^{p}\,e^{-m^{2}T}=(m^{2})^{\frac{D}{2}-p}\,\Gamma\left(p-\frac{D}{2}\right)\
,$ (4.2)

where $p=2,3$ correspond to our cases of interest $D=4,6$, respectively. Now,
using dimensional regularization, namely taking $D=2p-2\epsilon$ and expanding
the gamma function, we see the appearance of the usual $\frac{1}{\epsilon}$
pole as the leading divergent term: it corresponds precisely to the
logarithmic divergences seen in dimensional regularization [46].

In general, one has to deal also with IR divergences: for the sake of our
discussion, they can be avoided either by introducing an upper cutoff in the
proper time or by keeping a “small” mass regulator $m$, as we have done above.

###  4.1 Pure gravity in four dimensions

It has long been known since the pioneering work of ’t Hooft and Veltman [1],
that pure gravity with vanishing cosmological constant is a renormalizable
theory at one-loop in $D=4$. It is free of logarithmic divergences, while
other divergences are not seen in dimensional regularization, and in any case
they can be eliminated by renormalization. The same does not hold in the case
of a non-vanishing cosmological constant, as found by Christensen and Duff
[6]. Let us briefly review these statements in the light of our calculations.
Setting $D=4$ in (4.1) we see that the different powers of $T$ give rise to
the quartic, quadratic, and logarithmic divergences parametrized by $a_{0}$,
$a_{1}$ and $a_{2}$, respectively. In dimensional regularization only the
logarithmic divergences are visible. From (2.31) we read

$\left.a_{2}\right|_{D=4}=-\frac{29}{40}\,R^{2}+\frac{53}{45}\,R_{\mu\nu\rho\sigma}^{2}\
.$ (4.3)

These numerical values for the one-loop four-dimensional logarithmic
divergences of quantum gravity with non-vanishing cosmological constant
coincide precisely with those calculated long ago by Christensen and
Duff.777For comparison, one has to make evident the cosmological constant term
with the on-shell condition (2.1), which reads $R=4\Lambda$. The term
proportional to $R_{\mu\nu\rho\sigma}^{2}$ could be neglected, as thanks to
the Gauss-Bonnet theorem for four-dimensional Einstein manifolds it is
proportional to a total derivative, and thus eliminable from the effective
action, but the remaining term proportional to $R^{2}$ cannot be renormalized
away by redefining the parameters of the Einstein-Hilbert action. The theory
is not renormalizable.

On the other hand, setting the cosmological constant to vanish, one finds that
the on-shell background satisfies $R_{\mu\nu}=0$, and thus $R=0$. The
logarithmic divergence reduces to

$a_{2}\Big{|}_{\begin{subarray}{c}D=4\\\\[0.70004pt]
\Lambda=0\end{subarray}}=\frac{53}{45}\,R_{\mu\nu\rho\sigma}^{2}$ (4.4)

which in four dimensions is a total derivative, as discussed earlier, and can
be eliminated from the effective action. Thus, one recovers the result that
the one-loop logarithmic divergences of pure quantum gravity without
cosmological constant vanish in four dimensions. This property does not hold
true anymore at two-loops, as found by Goroff and Sagnotti [4] and verified by
van de Ven [5]. Returning to the one-loop divergences for vanishing
cosmological constant, one finds that also $a_{1}$ vanishes. This leaves only
the quartic divergence proportional to $a_{0}$, which gives the number of
degrees of freedom of the graviton. It requires a renormalization of the
cosmological constant back to zero, which makes the theory rather unnatural in
the technical sense of ’t Hooft [47], but in any case renormalizable at one-
loop.

For arbitrary nonzero values of the cosmological constant, the quadratic
divergence related to $a_{1}$ is not vanishing anymore and its value at $D=4$

$\left.a_{1}\right|_{D=4}=-\frac{8}{3}\,R$ (4.5)

reproduces the gauge-invariant result already computed in [30, 48]. It can be
renormalized away by redefining the Newton constant. Finally, the coefficient
$a_{3}$ gives rise to a finite term in the four-dimensional effective action,
but its physical meaning is unclear. It is gauge invariant, but infrared
divergences invalidate a local expansion of the effective action as delivered
by the small proper time approximation of the heat kernel, which is only
useful to locate the UV divergences, as far as we know. It might however
signal some property of quantum gravity which we are unaware of.

###  4.2 Pure gravity in six dimensions

The newly computed coefficient $a_{3}$ (2.32), allows us to see what happens
in six spacetime dimensions. Setting $D=6$ in the coefficients (2.29)–(2.32)
we find

$\displaystyle\begin{split}\left.a_{0}\right|_{D=6}&=9\;,\qquad\qquad\quad\left.a_{1}\right|_{D=6}=-\frac{3}{2}\;R\;,\qquad\qquad\quad\left.a_{2}\right|_{D=6}=-\frac{11}{20}\;R^{2}+\frac{21}{20}\;R_{\mu\nu\rho\sigma}^{2}\end{split}$
(4.6)
$\displaystyle\begin{split}\left.a_{3}\right|_{D=6}&=-\frac{799}{11340}\;R^{3}+\frac{481}{1680}\;R\,R_{\mu\nu\rho\sigma}^{2}-\frac{991}{5040}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}-\frac{71}{180}\;R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\
,\end{split}$ (4.7)

that furnish the full list of one-loop divergences of quantum gravity with
cosmological constant in six dimensions. We stress that these coefficients are
gauge invariant, and thus any other method of calculation should reproduce
these values.

A comparison with the literature can be made by setting the cosmological
constant to zero and considering the logarithmic divergence parametrized by
$a_{3}$, which reduces to

$a_{3}\Big{|}_{\begin{subarray}{c}D=6\\\\[0.70004pt]
\Lambda=0\end{subarray}}=-\frac{991}{5040}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}-\frac{71}{180}\,R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\
.$ (4.8)

These two remaining terms are proportional to two invariants that are
generally independent of each other. However, it turns out that in six
dimensions there exists an integral relation that connects them. It involves
the use of the Gauss-Bonnet theorem and the introduction of the Euler
character $\chi_{\rm E}(\mathcal{M})$, as explained in [49] and discussed more
extensively in appendix D. Bottom line, we can further simplify the
coefficient $a_{3}$, which becomes

$a_{3}\Big{|}_{\begin{subarray}{c}D=6\\\\[0.70004pt]
\Lambda=0\end{subarray}}=-\frac{9}{15120}\;R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}\;.$
(4.9)

It encodes the one-loop logarithmic divergences of pure gravity in six
dimensions. We are now in the position of carrying out a comparison with the
literature: our value in (4.9) is in complete agreement with van
Nieuwenhuizen’s pioneering calculation [2], besides a computational error in
the numerical factor in his equation (81), already noted a year later by
Critchley [3]. Furthermore, confirmation of this value is also found in more
recent works, see for instance [50, 51].

As for the general case of arbitrary cosmological constant, we are not aware
of similar calculations, although they would certainly be interesting to
pursue to further verify our findings.

##  5 Conclusions

In this work, we have investigated the computation of counterterms necessary
for the renormalization of the one-loop effective action of quantum gravity
with cosmological constant in arbitrary dimensions. Our results are complete
for dimensions $D<8$. The counterterms have been computed on-shell, so they
furnish gauge invariant quantities characteristic of the quantum theory of
gravity.

Our main contribution was the determination of the Seleey-DeWitt coefficient
$a_{3}(D)$ of perturbative quantum gravity, which to our knowledge has never
been reported in its full generality in the literature. When restricted to six
dimensions, it parameterizes the logarithmic divergence which was previously
known only for the case of vanishing cosmological constant. To cross-check our
calculations, we have used two distinct methods, a first-quantized description
of the graviton in terms of the ${\cal N}=4$ spinning particle and the time-
honored heat kernel method, finding complete agreement.

While the utility of heat kernel methods is well-known and they keep being
employed in many contexts, see for example [52, 53] for some recent
applications to trace anomalies, first-quantized methods for treating the
graviton with the ${\cal N}=4$ spinning particle are more recent and we have
championed them here to show their usefulness. In this respect, it would be
interesting to extend the present analysis to the first quantized model that
describes the ${\cal N}=0$ supergravity [54], i.e. the particle theory that
has in its spectrum the graviton, the dilaton, and the antisymmetric tensor
$B_{\mu\nu}$, as well as extend the present methods to the U($N$) spinning
particles [55, 56, 57, 58] to find a useful first-quantized way of describing
gravitational theories on complex (Kähler) manifolds, and finally also
consider double copy features on the worldline [59] to address gravitational
aspects from a different perspective.

## Acknowledgments

F.F. would like to thank the Galileo Galilei Institute (Firenze, Italy) for
its hospitality during part of this work. Our diagrams have been produced with
the help of TikZ-Feynman [60].

##  Appendix A Basis of invariants on Einstein manifolds

We use the following conventions for the curvature tensors:

$[\nabla_{\mu},\nabla_{\nu}]V^{\lambda}=R_{\mu\nu}{}^{\lambda}{}_{\rho}V^{\rho}\
,\ \ \ R_{\mu\nu}=R_{\lambda\mu}{}^{\lambda}{}_{\nu}\ ,\ \ \
R=R^{\mu}{}_{\mu}>0\ {\rm on\ spheres.}$ (A.1)

A $D$-dimensional Riemannian manifold without boundaries can be described
through an (infinite) basis of curvature monomials $\mathcal{K}_{i}^{n}$.
These are geometric invariants of order $n$ in the Riemann tensor, Ricci
tensor, and scalar curvature, with two covariant derivatives counting as a
Riemann tensor. They have been introduced by [61] and recently reviewed in
[62, 63]. At order $n=3$ we use the basis considered in [64] which is made of
$17$ independent invariants:

$\displaystyle\begin{array}[]{lll}\mathcal{K}_{1}=R^{3}&\mathcal{K}_{2}=RR_{\mu\nu}^{2}&\mathcal{K}_{3}=RR_{\mu\nu\rho\sigma}^{2}\\\\[8.53581pt]
\mathcal{K}_{4}=R_{\mu}{}^{\rho}R_{\rho}{}^{\nu}R_{\nu}{}^{\mu}&\mathcal{K}_{5}=R_{\mu\nu}R_{\rho\sigma}R^{\rho\mu\nu\sigma}&\mathcal{K}_{6}=R_{\mu\nu}R^{\mu\rho\sigma\lambda}R^{\nu}{}_{\rho\sigma\lambda}\\\\[8.53581pt]
\mathcal{K}_{7}=R_{\mu\nu}{}^{\rho\sigma}R_{\rho\sigma}{}^{\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}&\mathcal{K}_{8}=R_{\mu\rho\sigma\nu}R^{\rho\alpha\beta\sigma}R_{\alpha}{}^{\mu\nu}{}_{\beta}&\mathcal{K}_{9}=R\nabla^{2}R\\\\[8.53581pt]
\mathcal{K}_{10}=R_{\mu\nu}\nabla^{2}R^{\mu\nu}&\mathcal{K}_{11}=R_{\mu\nu\rho\sigma}\nabla^{2}R^{\mu\nu\rho\sigma}&\mathcal{K}_{12}=R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}R\\\\[8.53581pt]
\mathcal{K}_{13}=(\nabla_{\mu}R_{\rho\sigma})^{2}&\mathcal{K}_{14}=\nabla_{\mu}R_{\nu\rho}\nabla^{\nu}R^{\mu\rho}&\mathcal{K}_{15}=(\nabla_{\alpha}R_{\mu\nu\rho\sigma})^{2}\\\\[8.53581pt]
\mathcal{K}_{16}=\nabla^{2}R^{2}&\mathcal{K}_{17}=\nabla^{4}R\ .&\end{array}$
(A.8)

All other terms cubic in the curvature are linear combinations of the above
invariants after taking into account the symmetry properties and the Bianchi
identities of the Riemann tensor.

On Einstein manifolds, the basis (LABEL:R3) can be reduced further. Einstein
metrics are defined by the equation

$R_{\mu\nu}=\lambda g_{\mu\nu}$ (A.9)

that upon contraction leads to $R=\lambda D$. From the second Bianchi identity
one finds that $R$ and $\lambda$ are constant for $D>2$

$\nabla^{\mu}R_{\mu\nu}=\frac{1}{2}\nabla_{\nu}R\quad\longrightarrow\quad(D-2)\nabla_{\nu}\lambda=0\quad\longrightarrow\quad\nabla_{\nu}R=0$
(A.10)

and then from (A.9), one finds that the Ricci tensor is also covariantly
constant for $D>2$

$\nabla_{\alpha}R_{\mu\nu}=0\,.$ (A.11)

Let us consider now the second Bianchi identity, namely

$\nabla_{\rho}R_{\mu\nu\alpha\beta}+\nabla_{\beta}R_{\mu\nu\rho\alpha}+\nabla_{\alpha}R_{\mu\nu\beta\rho}=0\,,$
(A.12)

by contracting the above identity with $g^{\rho\mu}$ and using (A.11) one gets

$\nabla^{\mu}R_{\mu\nu\alpha\beta}=0\ .$ (A.13)

We are now in the position to reduce the six-dimensional basis of invariants
(LABEL:R3) on Einstein manifolds to a minimal set of independent ones, namely

$\mathcal{E}_{1}=R^{3}\;,\hskip
28.45274pt\mathcal{E}_{2}=R\,R_{\mu\nu\rho\sigma}^{2}\;,\hskip
28.45274pt\mathcal{E}_{3}=R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}\;,\hskip
28.45274pt\mathcal{E}_{4}=R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\;.$
(A.14)

Indeed we have

$\mathcal{K}_{9}=\mathcal{K}_{10}=0\
,\qquad\mathcal{K}_{2}=\frac{1}{D}\mathcal{E}_{1}\
,\qquad\mathcal{K}_{4}=-\mathcal{K}_{5}=\frac{1}{D^{2}}\mathcal{E}_{1}\
,\qquad\mathcal{K}_{6}=\frac{1}{D}\mathcal{E}_{2}\ .$ (A.15)

Moreover, the only term of (LABEL:R3) containing covariant derivatives and
non-vanishing on Einstein manifolds, i.e. $\mathcal{K}_{11}$,888Since, up to a
total derivative term, we have $\mathcal{K}_{15}=-\mathcal{K}_{11}$. can be
written as

$\displaystyle R_{\mu\nu\alpha\beta}\nabla^{2}R^{\mu\nu\alpha\beta}$
$\displaystyle=-R_{\mu\nu\alpha\beta}\nabla_{\rho}\left(\nabla^{\beta}R^{\mu\nu\rho\alpha}+\nabla^{\alpha}R^{\mu\nu\beta\rho}\right)=-2R_{\mu\nu\alpha\beta}\nabla_{\rho}\nabla^{\beta}R^{\mu\nu\rho\alpha}$
$\displaystyle=-2R_{\mu\nu\alpha\beta}\left(\nabla^{\beta}\nabla_{\rho}R^{\rho\alpha\mu\nu}+R^{\beta}{}_{\lambda}R^{\lambda\alpha\mu\nu}+R_{\rho}{}^{\beta\alpha}{}_{\lambda}R^{\rho\lambda\mu\nu}+R_{\rho}{}^{\beta\mu}{}_{\lambda}R^{\rho\alpha\lambda\nu}+R_{\rho}{}^{\beta\nu}{}_{\lambda}R^{\rho\alpha\mu\lambda}\right)$
$\displaystyle=-2R_{\mu\nu\alpha\beta}R_{\rho}{}^{\beta\alpha}{}_{\lambda}R^{\rho\lambda\mu\nu}+\frac{2}{D}R\,R_{\mu\nu\alpha\lambda}^{2}-4R_{\mu\nu\alpha\beta}R_{\rho}{}^{\beta\mu}{}_{\lambda}R^{\rho\alpha\lambda\nu}$
$\displaystyle=-R_{\mu\nu\alpha\beta}R^{\alpha\beta}{}_{\rho\lambda}R^{\rho\lambda\mu\nu}+\frac{2}{D}RR_{\mu\nu\alpha\beta}^{2}+4R_{\mu\nu\alpha\beta}R^{\nu\lambda\rho\alpha}R_{\lambda}{}^{\mu\beta}{}_{\rho}$
$\displaystyle=\frac{2}{D}\mathcal{E}_{2}-\mathcal{E}_{3}+4\mathcal{E}_{4}\ ,$
(A.16)

where we made use of the second Bianchi identity, antisymmetry of the Riemann
tensor and

$[\nabla_{\alpha},\nabla_{\beta}]R_{\mu\nu\rho\sigma}=R_{\alpha\beta\mu}{}^{\lambda}R_{\lambda\nu\rho\sigma}+R_{\alpha\beta\nu}{}^{\lambda}R_{\mu\lambda\rho\sigma}+R_{\alpha\beta\rho}{}^{\lambda}R_{\mu\nu\lambda\sigma}+R_{\alpha\beta\sigma}{}^{\lambda}R_{\mu\nu\rho\lambda}\
.$ (A.17)

Finally, let us discuss how the basis (A.14) further simplifies on maximally
symmetric spaces (MSS), which form a subset of the Einstein ones, where the
Riemann tensor is given by

$R_{\mu\nu\alpha\beta}=\frac{R}{D(D-1)}\left(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha}\right)\
.$ (A.18)

The invariants previously defined in (A.14) all collaps to $\mathcal{E}_{1}$,
and one finds

$\mathcal{E}_{2}=\frac{2}{D(D-1)}\mathcal{E}_{1}\
,\qquad\mathcal{E}_{3}=\frac{4}{D^{2}(D-1)^{2}}\mathcal{E}_{1}\
,\qquad\mathcal{E}_{4}=-\frac{D-2}{D^{2}(D-1)^{2}}\mathcal{E}_{1}\ .$ (A.19)

These relations allow us to evaluate the newly-computed coefficient $a_{3}(D)$
on MSS, giving (2.34) as a result.

##  Appendix B Worldline computations

###  B.1 Worldline propagators

The worldline propagators for the $\mathcal{N}=4$ spinning particle descend
from the free action (2.15). Regarding the bosonic quantum fluctuations
$q^{\mu}(\tau)$, we considered DBC during the evaluation of the derivative
expansion of the effective action, namely $q^{\mu}(0)=q^{\mu}(1)=0$. This
leads to the DBC worldline propagator defined by the two-point function

$\langle q^{\mu}(\tau)q^{\nu}(\sigma)\rangle=-2Tg^{\mu\nu}(x_{0})\Delta_{{\rm
D}}(\tau,\sigma)\ ,$ (B.1)

where

$\Delta_{{\rm
D}}(\tau,\sigma)=\left(\tau-1\right)\sigma\,\theta\left(\tau-\sigma\right)+\left(\sigma-1\right)\tau\,\theta\left(\sigma-\tau\right)\;.$
(B.2)

We also list the derivatives of the DBC propagator

$\displaystyle{}^{\bullet}\\!\Delta_{\rm D}(\tau,\sigma)$
$\displaystyle=\sigma-\theta(\sigma-\tau)$ (B.3)
$\displaystyle\Delta^{\hskip-1.42262pt\bullet}_{\rm D}(\tau,\sigma)$
$\displaystyle=\tau-\theta(\tau-\sigma)$ (B.4)
$\displaystyle{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{\rm
D}(\tau,\sigma)$ $\displaystyle=1-\delta(\tau-\sigma)$ (B.5)
$\displaystyle{}^{\bullet\bullet}\\!\Delta_{\rm D}(\tau,\sigma)$
$\displaystyle=\delta(\tau-\sigma)\;.$ (B.6)

Concerning the fermionic fields $\psi^{a}_{i}(\tau)$, we list here their
propagator

$\langle\psi^{a}_{i}(\sigma)\bar{\psi}^{b}_{j}(\tau)\rangle=\delta^{ab}\Delta_{\rm{AF}}\,{}_{ij}(\tau,\sigma)=\delta^{ab}\left(\begin{array}[]{cc}F(z,\tau,\sigma)&0\\\
0&F(\omega,\tau,\sigma)\\\ \end{array}\right)$ (B.7)

where each entry in the matrix is a ${\cal N}=2$ fermionic propagator [35]
defined as

$\displaystyle F(z,\tau,\sigma)$
$\displaystyle=z^{-(\tau-\sigma)}\left(\frac{1}{z+1}\right)\left(z\,\theta(\tau-\sigma)-\theta(\sigma-\tau)\right)$
(B.8) $\displaystyle F(\omega,\tau,\sigma)$
$\displaystyle=\omega^{-(\tau-\sigma)}\left(\frac{1}{\omega+1}\right)\left(\omega\,\theta(\tau-\sigma)-\theta(\sigma-\tau)\right)\
.$ (B.9)

Next, we have the ghost variable propagators, defined as

$\displaystyle\langle a^{\mu}(\tau)a^{\nu}(\sigma)\rangle$
$\displaystyle=2Tg^{\mu\nu}(x_{0})\Delta_{{}_{\rm
gh}}(\tau,\sigma)=2Tg^{\mu\nu}(x_{0})\delta(\tau-\sigma)$ (B.10)
$\displaystyle\langle b^{\mu}(\tau)c^{\nu}(\sigma)\rangle$
$\displaystyle=-4Tg^{\mu\nu}(x_{0})\Delta_{{}_{\rm
gh}}(\tau,\sigma)=-4Tg^{\mu\nu}(x_{0})\delta(\tau-\sigma)\;.$ (B.11)

As previously stated, the calculation involving these propagators may result
in products/derivatives of delta distributions, which are ill-defined, but
also divergent quantities such as $\delta(\tau,\tau)$, therefore one needs to
regularize the path integral. In the present work, we chose worldline
dimensional regularization (DR), which consists in continuing the compact time
direction with the addition of $d$ non-compact extra dimensions, i.e.
extending the space $\tau\in[0,1]\rightarrow
t^{\alpha}=\left(\tau,\mathbf{t}\right)\in[0,1]\times\mathbb{R}^{n}$. More
details can be found for example in [28]; here we list the dimensional
regularized expression of propagators, exploited during intermediate steps
when performing computations. The $d+1$ extended propagators read

$\displaystyle\Delta_{\rm D}(t,s)$
$\displaystyle=\int\frac{d^{d}k}{(2\pi)^{d}}\sum_{m=1}^{\infty}\frac{-2}{(\pi
m)^{2}+\mathbf{k}^{2}}\sin(\pi m\tau)\sin(\pi
m\sigma)\,e^{i\mathbf{k}\cdot(\mathbf{t}-\mathbf{s})}$ (B.12)
$\displaystyle\Delta_{\rm gh}(t,s)$
$\displaystyle=\int\frac{d^{d}k}{(2\pi)^{d}}\sum_{m=1}^{\infty}2\sin(\pi
m\tau)\sin(\pi
m\sigma)e^{i\mathbf{k}\cdot(\mathbf{t}-\mathbf{s})}=\delta(\tau-\sigma)\delta^{d}(t-s)$
(B.13) $\displaystyle F(\theta,t,s)$
$\displaystyle=-i\int\frac{d^{d}k}{(2\pi)^{d}}\sum_{r\in
Z+\frac{1}{2}}\frac{2\pi r\gamma^{0}+k\cdot\gamma-\theta}{(2\pi
r)^{2}+\mathbf{k}^{2}-\theta^{2}}\,e^{2\pi
ir(\tau-\sigma)}e^{i\mathbf{k}\cdot(\mathbf{t}-\mathbf{s})}$ (B.14)

where in the dimensional regularized expressions one has $\mathbf{t}$ as
$d$-dimensional vector and $\gamma^{\alpha}$ are the Dirac matrices in the
$(d+1)$-dimensional extended space. Each extended worldline propagator
satisfies a generalization of its own one-dimensional Green equation

$\displaystyle\partial_{\mu}\partial^{\mu}\Delta_{\rm D}(t,s)$
$\displaystyle=\delta^{d+1}(t-s)$ (B.15)
$\displaystyle\left(\overset{\rightarrow}{\not{\partial}}+i\theta\right)F(\theta,t,s)$
$\displaystyle=\delta_{\rm AF}(\tau-\sigma)\delta^{d}(t-s)$ (B.16)
$\displaystyle
F(\theta,t,s)\left(-\overset{\leftarrow}{\not{\partial}}+i\theta\right)$
$\displaystyle=\delta_{\rm AF}(\tau-\sigma)\delta^{d}(t-s)\ ,$ (B.17)

where $\delta_{\rm AF}$ is the delta distribution acting on antiperiodic
functions on $[0,1]$ and where a slashed derivative is the usual contraction
between derivative and gamma matrices. For computational purposes, the index
contractions in $d+1$ dimensions serve mostly as a bookkeeping device to keep
track of which derivative can be contracted to which vertex to produce the
$(d+1)$-dimensional delta function. The delta functions in (B.15)–(B.17) are
only to be used in $d+1$ dimensions; then, by using partial integration one
casts the various loop integrals in a form that can be computed by sending
$d\to 0$ first. At this stage, one can use the one-dimensional propagators
(B.2)–(B.7), and $\gamma^{0}=1$ (with no extra factors arising from the Dirac
algebra in $d+1$ dimensions).

###  B.2 Analisys of perturbative contributions

In this appendix, we will give the essential details on the evaluation of the
path integral average and its subsequent modular integration (2.17), which
produces the Seleey-DeWitt coefficients to be inserted in (2.18). As
anticipated in Section 2.3, in order to find all the possible contributions to
order $T^{3}$ we have to expand the exponential with the interacting action
(2.20), written more compactly as in (2.21) which in particular carries the
following vertices

$\displaystyle S_{\rm K1}=\int
d\tau\;\frac{1}{3}R_{\alpha\mu\nu\beta}\,q^{\alpha}q^{\beta}\left(\dot{q}^{\mu}\dot{q}^{\nu}+\text{\rm
gh}\right)\hskip 5.69046pt;\hskip 5.69046ptS_{\rm C1}=\int
d\tau\;\frac{1}{2}R_{\alpha\mu
ab}\,\dot{q}^{\mu}q^{\alpha}\,\bar{\psi}^{a}\cdot\psi^{b}$ (B.18)
$\displaystyle DS_{\rm C1}=\int d\tau\;\frac{1}{3}\nabla_{\alpha}R_{\beta\mu
ab}\,\dot{q}^{\mu}q^{\alpha}q^{\beta}\,\bar{\psi}^{a}\cdot\psi^{b}\hskip
5.69046pt;\hskip 5.69046ptD^{2}S_{\rm C1}=\int
d\tau\;\frac{1}{8}\nabla_{\alpha}\nabla_{\beta}R_{\gamma\mu
ab}\,\dot{q}^{\mu}q^{\alpha}q^{\beta}q^{\gamma}\,\bar{\psi}^{a}\cdot\psi^{b}$
(B.19) $\displaystyle S_{\rm C2}=\int
d\tau\;\frac{1}{24}R^{\tau}{}_{\alpha\beta\mu}R_{\gamma\tau
ab}\,\dot{q}^{\mu}q^{\alpha}q^{\beta}q^{\gamma}\,\bar{\psi}^{a}\cdot\psi^{b}\hskip
5.69046pt;\hskip 5.69046ptS_{\rm F}=\int
d\tau\;R_{abcd}\,\bar{\psi}^{a}\cdot\psi^{b}\bar{\psi}^{c}\cdot\psi^{d}$
(B.20) $\displaystyle DS_{\rm F}=\int
d\tau\;q^{\alpha}\nabla_{\alpha}R_{abcd}\,\bar{\psi}^{a}\cdot\psi^{b}\bar{\psi}^{c}\cdot\psi^{d}\hskip
5.69046pt;\hskip 5.69046ptD^{2}S_{\rm F}=\int
d\tau\;\frac{1}{2}q^{\alpha}q^{\beta}\nabla_{\alpha}\nabla_{\beta}R_{abcd}\,\bar{\psi}^{a}\cdot\psi^{b}\bar{\psi}^{c}\cdot\psi^{d}$
(B.21) $\displaystyle S_{\rm V}=-\int d\tau\;\Omega R\ .$ (B.22)

Through Wick contractions they result in a plethora of different terms, most
of which either give rise to disconnected diagrams (so that they are easily
taken care of afterward, as connected diagrams exponentiate and there is no
need to compute them anew) or can easily be shown to vanish. To start with, we
see that the vertex $S_{\rm V}$ in (B.22) does not depend on $q$, $\psi$ nor
on $\bar{\psi}$, thus it can be taken out of the path integral and it remains
exponentiated (as will be all connected worldline diagrams). Let’s see some
other illustrative examples. For instance, a disconnected diagram arises from
the term

$-\frac{1}{8}T\langle S_{\rm K1}S_{\rm F}^{2}\rangle\sim\leavevmode\hbox
to81.09pt{\vbox to41.26pt{\pgfpicture\makeatletter\hbox{\hskip
40.54518pt\lower-20.62825pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex[dot] (b) at
(0,0) {}; \vertex[dot] (e) at (1.8,0) {}; \vertex[dot](c)at(3.2,0){};
{}{{}}{}{}{{{{}{}{}{}}}{{{}{}{}{}}}{{{}{}{}{}}}{{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-10.99995pt}{-8.91698pt}{-19.91693pt}{-19.91693pt}{-19.91693pt}\pgfsys@curveto{-30.91689pt}{-19.91693pt}{-39.83386pt}{-10.99995pt}{-39.83386pt}{0.0pt}\pgfsys@curveto{-39.83386pt}{10.99995pt}{-30.91689pt}{19.91693pt}{-19.91693pt}{19.91693pt}\pgfsys@curveto{-8.91698pt}{19.91693pt}{0.0pt}{10.99995pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{}{{{{}{}{}{}}}
{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}{} {}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@curveto{39.83368pt}{10.99995pt}{30.9167pt}{19.91693pt}{19.91675pt}{19.91693pt}\pgfsys@curveto{8.9168pt}{19.91693pt}{-0.00018pt}{10.99995pt}{-0.00018pt}{0.0pt}\pgfsys@curveto{-0.00018pt}{-10.99995pt}{8.9168pt}{-19.91693pt}{19.91675pt}{-19.91693pt}\pgfsys@curveto{30.9167pt}{-19.91693pt}{39.83368pt}{-10.99995pt}{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\propag[fer] (e) to
[in=90, out=90,looseness=.5] (c); \propag[fer] (c) to [in=-90,
out=-90,looseness=.5] (e); \propag[fer] (c) to [in=90, out=90,looseness=1.5]
(e); \propag[fer] (e) to [in=-90, out=-90,looseness=1.5] (c); \vertex[ringdot]
(d) at (.2,-.5) {}; \vertex[ringdot] (d) at (-.2,.5) {};
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\sim\text{disconnected}\;.$
(B.23)

Other diagrams can be shown to be zero exploiting (anti)symmetry of the tensor
structures and/or of the resulting propagators, like

$\displaystyle-\frac{1}{32T^{2}}\langle S_{\rm K1}^{2}S_{\rm C1}\rangle$
$\displaystyle\sim R_{\mu\nu ab}\langle\bar{\psi}^{a}\cdot\psi^{b}\rangle=0$
(B.24) $\displaystyle\frac{1}{4}\langle S_{\rm K1}S_{\rm C1}S_{\rm F}\rangle$
$\displaystyle\sim R_{ab}R_{\mu\nu}{}^{ab}R_{\alpha\rho\sigma\beta}=0\ .$
(B.25)

Lastly, others are simply zero after explicitly calculating the integrals, as
happens for $\langle S_{\rm C2}S_{\rm F}\rangle$.

We report now the surviving contributions (2.22) in full glory:

$\displaystyle\langle e^{-S_{\rm int}}\big{\rangle}\Big{|}_{T^{3}}$
$\displaystyle=\langle S_{\rm KIN}\rangle$ (A)
$\displaystyle\phantom{=}-\frac{1}{96T}R_{\alpha\mu\nu\beta}R_{\gamma\lambda
ab}R_{\delta\epsilon
cd}\int_{012}\langle\dot{q}^{\mu}_{0}\dot{q}^{\nu}_{0}q^{\alpha}_{0}q^{\beta}_{0}\dot{q}^{\lambda}_{1}q^{\gamma}_{1}\dot{q}^{\epsilon}_{2}q^{\delta}_{2}\rangle\langle\bar{\psi}^{a}_{1}\cdot\psi_{1}^{b}\bar{\psi}^{c}_{2}\cdot\psi_{2}^{d}\rangle$
(B) $\displaystyle\phantom{=}+\frac{1}{48}R_{\alpha\mu
ab}R^{\tau}{}_{\beta\lambda\nu}R_{\rho\tau
cd}\int_{01}\langle\dot{q}^{\mu}_{0}q^{\alpha}_{0}\dot{q}^{\nu}_{1}q^{\beta}_{1}q^{\lambda}_{1}q^{\rho}_{1}\rangle\langle\bar{\psi}^{a}_{0}\cdot\psi_{0}^{b}\bar{\psi}^{c}_{1}\cdot\psi_{1}^{d}\rangle$
(C) $\displaystyle\begin{aligned}
\hfil\displaystyle\begin{split}\phantom{=}+\frac{1}{18}\nabla_{\alpha}R_{\beta\mu
ab}\nabla_{\lambda}R_{\rho\nu
cd}\int_{01}\langle\dot{q}^{\mu}_{0}q^{\alpha}_{0}q^{\beta}_{0}\dot{q}^{\nu}_{1}q^{\lambda}_{1}q^{\rho}_{1}\rangle\,\langle\bar{\psi}^{a}_{0}\cdot\psi^{b}_{0}\bar{\psi}^{c}_{1}\cdot\psi^{d}_{1}\rangle\\\
\phantom{=}+\frac{1}{16}\nabla_{\alpha}\nabla_{\beta}R_{\lambda\mu
ab}R_{\rho\nu
cd}\int_{01}\langle\dot{q}^{\mu}_{0}q^{\alpha}_{0}q^{\beta}_{0}q^{\lambda}_{0}\dot{q}^{\nu}_{1}q^{\rho}_{1}\rangle\,\langle\bar{\psi}^{a}_{0}\cdot\psi^{b}_{0}\bar{\psi}^{c}_{1}\cdot\psi^{d}_{1}\rangle\end{split}\end{aligned}$
(D) $\displaystyle\phantom{=}+\frac{T}{8}R_{\alpha\mu ab}R_{\beta\nu
cd}R_{efgh}\int_{012}\langle\dot{q}^{\mu}_{0}\dot{q}^{\nu}_{1}q^{\alpha}_{0}q^{\beta}_{1}\rangle\langle\bar{\psi}^{a}_{0}\cdot\psi^{b}_{0}\bar{\psi}^{c}_{1}\cdot\psi^{d}_{1}\psi^{e}_{2}\cdot\bar{\psi}^{f}_{2}\psi^{g}_{2}\cdot\bar{\psi}^{h}_{2}\rangle$
(E)
$\displaystyle\phantom{=}+\frac{1}{6}R_{abcd}R_{efgh}R_{lmno}\int_{012}\langle\bar{\psi}^{a}_{0}\cdot\psi^{b}_{0}\,\bar{\psi}^{c}_{0}\cdot\psi^{d}_{0}\bar{\psi}^{e}_{1}\cdot\psi^{f}_{1}\,\bar{\psi}^{g}_{1}\cdot\psi^{h}_{1}\bar{\psi}^{l}_{2}\cdot\psi^{m}_{2}\,\bar{\psi}^{n}_{2}\cdot\psi^{o}_{2}\rangle$
(F) $\displaystyle\begin{aligned}
\hfil\displaystyle\begin{split}&\phantom{=}+\frac{T^{2}}{2}\nabla_{\alpha}R_{abcd}\nabla_{\beta}R_{efgh}\int_{01}\langle\dot{q}^{\alpha}_{0}\dot{q}^{\beta}_{1}\rangle\langle\bar{\psi}^{a}_{0}\cdot\psi^{b}_{0}\,\bar{\psi}^{c}_{0}\cdot\psi^{d}_{0}\bar{\psi}^{e}_{1}\cdot\psi^{f}_{1}\,\bar{\psi}^{g}_{1}\cdot\psi^{h}_{1}\rangle\\\
&\phantom{=}+\frac{T^{2}}{2}\nabla_{\alpha}\nabla_{\beta}R_{abcd}R_{efgh}\int_{01}\langle\dot{q}^{\alpha}_{0}\dot{q}^{\beta}_{0}\rangle\langle\bar{\psi}^{a}_{0}\cdot\psi^{b}_{0}\,\bar{\psi}^{c}_{0}\cdot\psi^{d}_{0}\bar{\psi}^{e}_{1}\cdot\psi^{f}_{1}\,\bar{\psi}^{g}_{1}\cdot\psi^{h}_{1}\rangle\
.\end{split}\end{aligned}$ (G)

We decided to keep both the ghost fields and the term $S_{\rm KIN}$ implicit
so as not to further burden the notation, and we combined the two terms of
pure spin connection with covariant derivatives $\frac{1}{2}DS_{\rm
C1}^{2}+D^{2}S_{\rm C1}S_{\rm C1}$ in eq. (D) and the two terms of the Taylor
expansion of the four-fermions action $\frac{1}{2}T^{2}DS_{\rm
F}^{2}+T^{2}D^{2}S_{\rm F}S_{\rm F}$ in eq. (G), for reasons that will become
clear shortly thereafter. Below we illustrate the main steps of the
calculation for each contribution.

  1. (A)

The first contribution we called $S_{\rm KIN}$ to indicate economically the
sum of all terms arising _only_ from the pure kinetic part of the interacting
action. In the notation previously presented it would read

$S_{\rm KIN}=-\frac{1}{384T^{3}}S_{\rm K1}^{3}+\frac{1}{32T^{2}}DS_{\rm
K1}^{2}+\frac{S_{\rm K1}}{16T^{2}}\left(S_{\rm K2}+D^{2}S_{\rm
K1}\right)-\frac{1}{4T}S_{\rm K3}\ .$ (B.26)

It has already been computed in [31] (eq. 20) and can be read out, translated
in our basis (A.14), from the exponential of connected diagrams as

$\langle S_{\rm
KIN}\rangle=\frac{T^{3}}{7!}\left(-\frac{16}{9}\frac{\mathcal{E}_{1}}{D^{2}}+\frac{2}{3}\frac{\mathcal{E}_{2}}{D}+\frac{17}{9}\mathcal{E}_{3}+\frac{28}{9}\mathcal{E}_{4}\right)\
.$ (B.27)

  2. (B)

Regarding the contribution obtained by coupling the bosonic kinetic term and
the spin connection, once the contractions have been evaluated, one can use
integration by parts (IBP) and DR to reduce integrals to a set of independent
ones in $z$, modulo their $\omega$ partner.999This should also be understood
for forthcoming worldline integrals when the fermionic integrands depend
_only_ on the Wilson variable $z$. We list here such integrals

$\displaystyle\int_{012}{}^{\bullet}\\!\Delta(\tau,\sigma)\Delta^{\hskip-1.42262pt\bullet}(\tau,\sigma){}^{\bullet}\\!\Delta(\tau,\rho)\Delta^{\hskip-1.42262pt\bullet}(\tau,\rho)F(z,\sigma,\rho)F(z,\rho,\sigma)=-\frac{z}{120\left(1+z\right)^{2}}$
(B.28)
$\displaystyle\int_{012}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}(\tau,\rho){}^{\bullet}\\!\Delta(\tau,\tau)\Delta^{\hskip-1.42262pt\bullet}(\tau,\sigma)\Delta(\sigma,\rho)F(z,\sigma,\rho)F(z,\rho,\sigma)=-\frac{1}{720}\frac{z}{(z+1)^{2}}\
,$ (B.29)

so that the final result for $-\frac{1}{8T}\langle S_{\rm K1}S^{2}_{\rm
SC1}\rangle$ reads

$T^{3}\frac{\mathcal{E}_{2}}{D}\left[-\frac{1}{90}\frac{z}{(z+1)^{2}}-\frac{1}{90}\frac{\omega}{(\omega+1)^{2}}\right]+T^{3}\mathcal{E}_{3}\left[-\frac{z}{60\left(1+z\right)^{2}}-\frac{\omega}{60\left(1+\omega\right)^{2}}\right]\
.$ (B.30)

  3. (C)

The contribution of pure spin connection has already been evaluated in Section
2.3; we report here the result for completeness:

$\langle S_{\rm C1}S_{\rm
C2}\rangle=\left(\frac{1}{180}\frac{\mathcal{E}_{2}}{D}+\frac{1}{120}\mathcal{E}_{3}\right)\left[\frac{z}{(z+1)^{2}}+\frac{\omega}{(\omega+1)^{2}}\right]\
.$ (B.31)

  4. (D)

The contribution of pure spin connection with covariant derivatives is made up
of the sum of two pieces: using Einstein manifolds simplifications one can
collect the whole result, with only the following diagrams to be evaluated

$\displaystyle\int_{01}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}(\sigma,\tau)\Delta(\sigma,\tau)\Delta(\sigma,\tau)F(z,\sigma,\tau)F(z,\tau,\sigma)=\frac{z}{45(z+1)^{2}}$
(B.32)
$\displaystyle\int_{01}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}(\sigma,\tau)\Delta(\sigma,\tau)\Delta(\sigma,\sigma)F(z,\sigma,\tau)F(z,\tau,\sigma)=\frac{z}{60(z+1)^{2}}\
,$ (B.33)

getting the final result

$\left\langle\frac{1}{2}DS_{\rm C1}^{2}+D^{2}S_{\rm C1}S_{\rm
C1}\right\rangle=-\frac{T^{3}}{360}\left[\frac{z}{(z+1)^{2}}+\frac{\omega}{(\omega+1)^{2}}\right]\left(2\frac{\mathcal{E}_{2}}{D}-\mathcal{E}_{3}+4\mathcal{E}_{4}\right)\
.$ (B.34)

  5. (E)

This is produced from the coupling of spin connection to the four-fermions
vertex. The list of independent diagrams one needs to evaluate is

$\displaystyle\int_{012}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}(\sigma,\tau)\Delta(\sigma,\tau)F(z,\rho,\tau)F(z,\tau,\rho)F(z,\rho,\sigma)F(z,\sigma,\rho)=-\frac{z^{2}}{12(z+1)^{4}}$
(B.35)
$\displaystyle\int_{012}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}(\sigma,\tau)\Delta(\sigma,\tau)F(z,\rho,\tau)F(z,\tau,\sigma)F(z,\sigma,u)F(z,\rho,\rho)=-\frac{z(z-1)^{2}}{48(z+1)^{4}}$
(B.36)
$\displaystyle\int_{012}{}^{\bullet}\\!\Delta(\sigma,\tau)\Delta^{\hskip-1.42262pt\bullet}(\sigma,\tau)F(z,\sigma,\rho)F(z,\rho,\sigma)F(\omega,\tau,\rho)F(\omega,u,\tau)=-\frac{z\omega}{12(z+1)^{2}(\omega+1)^{2}}\
,$ (B.37)

such that the final result reads

$\displaystyle\left\langle\frac{1}{2}T^{2}S_{\rm C1}^{2}S_{\rm
F}\right\rangle$
$\displaystyle=T^{3}\frac{\mathcal{E}_{2}}{D}\left[\frac{z\left(z-1\right)^{2}}{12\left(z+1\right)^{4}}+\frac{\omega\left(\omega-1\right)^{2}}{12\left(\omega+1\right)^{4}}\right]$
(B.38)
$\displaystyle\phantom{=}+T^{3}\mathcal{E}_{3}\left[\frac{z^{2}}{12\left(z+1\right)^{4}}+\frac{\omega^{2}}{12\left(\omega+1\right)^{4}}+\frac{z\omega}{3\left(z+1\right)^{2}\left(\omega+1\right)^{2}}\right]\
.$ (B.39)

  6. (F)

The one arising from the pure fermionic vertex is quite tricky to evaluate at
once. To simplify the calculation, it is possible to exploit the subtle double
copy structure underlying the $\mathcal{N}=4$ spinning particle, rewriting
this term as a sum of contributions coming from the two copies of the
$\mathcal{N}=2$ particles, i.e. for the two values of the internal index
$i=1,2$. This allows us to rewrite the above term as

$\displaystyle\frac{1}{6}S_{\rm F}^{3}$
$\displaystyle=\frac{1}{6}S_{1}^{3}(z)+\frac{1}{6}S_{1}^{3}(\omega)+\frac{1}{6}S_{\rm
mix}^{3}+\frac{1}{2}S_{1}(z)S_{2}^{2}(\omega)+\frac{1}{2}S_{1}(z)^{2}S_{2}(\omega)$
(B.40) $\displaystyle\phantom{=}+\frac{1}{2}S_{1}^{2}(z)S_{\rm
mix}+\frac{1}{2}S_{1}(z)S_{\rm mix}^{2}+\frac{1}{2}S_{2}^{2}(\omega)S_{\rm
mix}+\frac{1}{2}S_{2}(\omega)S_{\rm mix}^{2}+S_{1}(z)S_{2}(\omega)S_{\rm mix}\
.$ (B.41)

In the previous expression, we defined for simplicity the actions

$\displaystyle S_{1}(z)$ $\displaystyle=\int
d\tau\,TR_{abcd}\,\bar{\psi}^{1a}\psi_{1}^{b}\bar{\psi}^{1c}\psi_{1}\,^{d}$
(B.42) $\displaystyle S_{\rm mix}$ $\displaystyle=\int
d\tau\left(TR_{abcd}\,\bar{\psi}^{a1}\psi^{b}_{1}\bar{\psi}^{2c}\psi^{d}_{2}+1\leftrightarrow
2\right)\ ,$ (B.43)

where we explicated the flavor index. Once performed the contractions with
Mathematica, based on the _xTensor_ package [65], we further simplify the
result by reducing tensor structures using Bianchi identity as follows

$\displaystyle
R_{\alpha\beta}{}^{\rho\sigma}R^{\alpha\mu\beta\nu}R_{\rho\mu\sigma\nu}=\frac{1}{4}\mathcal{E}_{3}$
(B.44) $\displaystyle
R_{\mu\alpha\nu\beta}R^{\mu\rho\nu\sigma}R^{\alpha}{}_{\sigma}{}^{\beta}{}_{\rho}=-\frac{1}{4}\mathcal{E}_{3}+\mathcal{E}_{4}\
.$ (B.45)

Once evaluated the fermionic diagrams our final answer for
$\langle\frac{1}{6}T^{3}S_{\rm F}^{3}\rangle$ is

$\displaystyle\begin{split}&\phantom{=+}\frac{\mathcal{E}_{1}}{D^{2}}\left[-\frac{(-1+z)^{2}z(1+(-8+z)z)}{6(1+z)^{6}}-\frac{(-1+\omega)^{2}\omega(1+(-8+\omega)\omega)}{6(1+\omega)^{6}}\right]\\\\[3.00003pt]
&\phantom{=}+\frac{\mathcal{E}_{2}}{D}\left[-\frac{(z-1)^{2}z^{2}}{(z+1)^{6}}-\frac{(\omega-1)^{2}\omega^{2}}{(\omega+1)^{6}}-\frac{2(-1+z)^{2}z\omega}{(1+z)^{4}(1+\omega)^{2}}-\frac{2z(-1+\omega)^{2}\omega}{(1+z)^{2}(1+\omega)^{4}}\right]\\\\[3.00003pt]
&\phantom{=}+\mathcal{E}_{3}\left[\frac{z^{4}-2z^{3}+z^{2}}{6(z+1)^{6}}+\frac{\omega^{4}-2\omega^{3}+\omega^{2}}{6(\omega+1)^{6}}-\frac{2z^{2}\omega\
}{(1+z)^{4}(1+\omega)^{2}}-\frac{2z\omega^{2}}{(1+\
z)^{2}(1+\omega)^{4}}\right]\\\\[3.00003pt]
&\phantom{=}+\mathcal{E}_{4}\left[-\frac{4\omega^{3}}{3(\omega+1)^{6}}-\frac{4z^{3}}{3(z+1)^{6}}-\frac{4\omega
z}{3(\omega+1)^{2}(z+1)^{2}(\omega+1)^{2}}\right]\ .\end{split}$ (B.46)

  7. (G)

Finally, one has the term coming from the Taylor expansion of the four-
fermions interaction, made of two pieces. Just as in the (D) case, one can
collect them into a single one working on the tensorial structure and using
Einstein manifolds simplifications. The resulting diagrams to be evaluated are

$\displaystyle\int_{01}\Delta(\tau,\tau)F(z,\sigma,\tau)F(z,\sigma,\tau)F(z,\tau,\sigma)F(z,\tau,\sigma)=-\frac{z^{2}}{6(z+1)^{4}}$
(B.47)
$\displaystyle\int_{01}\Delta(\tau,\tau)F(z,\sigma,\tau)F(z,\tau,\sigma)F(\omega,\sigma,\tau)F(\omega,\tau,\sigma)=-\frac{\omega
z}{6(\omega+1)^{2}(z+1)^{2}}\ ,$ (B.48)

since the partner diagrams $\Delta(\tau,\tau)\rightarrow\Delta(\tau,\sigma)$
can easily be seen to be proportional to the above. The final answer for
$\langle\frac{1}{2}T^{2}DS_{\rm F}^{2}+T^{2}D^{2}S_{\rm F}S_{\rm F}\rangle$ is
then

$T^{3}\left[\frac{z^{2}}{12\left(z+1\right)^{4}}+\frac{\omega^{2}}{12\left(\omega+1\right)^{4}}+\frac{z\omega}{3\left(z+1\right)^{2}\left(\omega+1\right)^{2}}\right]\left(2\frac{\mathcal{E}_{2}}{D}-\mathcal{E}_{3}+4\mathcal{E}_{4}\right)\
.$ (B.49)

###  B.3 Summing up all the pieces

We can collect the result of all the computations above as follows

$\alpha_{3}(z,\omega,D)=c_{0}(z,\omega)\,\frac{\mathcal{E}_{1}}{D^{2}}+c_{1}(z,\omega)\,\frac{\mathcal{E}_{2}}{D}+c_{2}(z,\omega)\,\mathcal{E}_{3}+c_{3}(z,\omega)\,\mathcal{E}_{4}\
,$ (B.50)

with the coefficients

$\displaystyle\begin{split}c_{0}(z,\omega)&=-\frac{1}{2835}-\frac{z\left(z^{2}-8z+1\right)(z-1)^{2}}{6(z+1)^{6}}-\frac{(\omega-1)^{2}\omega\left(\omega^{2}-8\omega+1\right)}{6(\omega+1)^{6}}\end{split}$
(B.51)
$\displaystyle\begin{split}c_{1}(z,\omega)&=\frac{1}{7560}-\frac{z}{90(z+1)^{2}}-\frac{\omega}{90(\omega+1)^{2}}+\frac{z(z-1)^{2}}{12(z+1)^{4}}+\frac{(\omega-1)^{2}\omega}{12(\omega+1)^{4}}+\frac{z^{2}}{6(z+1)^{4}}+\frac{\omega^{2}}{6(\omega+1)^{4}}\\\\[3.00003pt]
&\phantom{=}-\frac{z^{2}(z-1)^{2}}{(z+1)^{6}}-\frac{(\omega-1)^{2}\omega^{2}}{(\omega+1)^{6}}-\frac{2\omega
z(z-1)^{2}}{(\omega+1)^{2}(z+1)^{4}}-\frac{2(\omega-1)^{2}\omega
z}{(\omega+1)^{4}(z+1)^{2}}+\frac{2z\omega}{3\left(z+1\right)^{2}\left(\omega+1\right)^{2}}\end{split}$
(B.52)
$\displaystyle\begin{split}c_{2}(z,\omega)&=\frac{17}{45360}-\frac{z}{180(z+1)^{2}}-\frac{\omega}{180(\omega+1)^{2}}+\frac{(z-1)^{2}z^{2}}{6(z+1)^{6}}+\frac{(\omega-1)^{2}\omega^{2}}{6(\omega+1)^{6}}\\\
&\phantom{=}-\frac{2\omega
z^{2}}{(\omega+1)^{2}(z+1)^{4}}-\frac{2\omega^{2}z}{(\omega+1)^{4}(z+1)^{2}}\end{split}$
(B.53)
$\displaystyle\begin{split}c_{3}(z,\omega)&=\frac{1}{1620}-\frac{z}{90(z+1)^{2}}-\frac{\omega}{90(\omega+1)^{2}}+\frac{z^{2}}{3\left(z+1\right)^{4}}+\frac{\omega^{2}}{3\left(\omega+1\right)^{4}}+-\frac{4\omega^{3}}{3(\omega+1)^{6}}-\frac{4z^{3}}{3(z+1)^{6}}\
.\end{split}$ (B.54)

At this point, in order to get the full $T^{3}$ correction, we first need to
recall the connected diagrams coming from the first- and second-order
expansion of the path integral, namely

$\displaystyle\alpha_{1}(z,\omega,D)$
$\displaystyle=\left(\frac{5}{12}+\Omega-\frac{z}{(z+1)^{2}}-\frac{\omega}{(\omega+1)^{2}}\right)R$
(B.55) $\displaystyle\alpha_{2}(z,\omega,D)$
$\displaystyle=\left(-\frac{1}{180}+\frac{1}{2}\left[\frac{z(z-1)^{2}}{(z+1)^{4}}+\frac{\omega(\omega-1)^{2}}{(\omega+1)^{4}}\right]\right)\frac{R^{2}}{D}$
$\displaystyle\phantom{=}+\left(\frac{1}{180}+\frac{1}{2}\left[\frac{\omega^{2}}{(\omega+1)^{4}}+\frac{z^{2}}{(z+1)^{4}}+\frac{4\omega
z}{(\omega+1)^{2}(z+1)^{2}}\right]-\frac{1}{12}\left[\frac{\omega}{(\omega+1)^{2}}+\frac{z}{(z+1)^{2}}\right]\right)R_{\mu\nu\rho\sigma}^{2}$
$\displaystyle\equiv\beta_{1}(z,\omega)\,\frac{R^{2}}{D}+\beta_{2}(z,\omega)\,R_{\mu\nu\rho\sigma}^{2}\
,$ (B.56)

which are in complete accordance with the results of [14, 15] and indeed
reproduce the $a_{1}$ and $a_{2}$ coefficients (2.30)–(2.31). Now we can move
to the exponentiation of all the connected diagrams, including the new
results. The expectation value $\big{\langle}e^{-S_{\rm int}}\big{\rangle}$ in
(2.17) can be compactly written as

$\displaystyle\begin{split}\Big{\langle}e^{-S_{\rm
int}}\Big{\rangle}&=\exp\Bigg{[}T\,\alpha_{1}\,R+T^{2}\,\left(\beta_{1}\,\frac{R^{2}}{D}+\beta_{2}\,R_{\mu\nu\rho\sigma}^{2}\right)\\\
&\phantom{=}+T^{3}\,\left(c_{0}\,\frac{R^{3}}{D^{2}}+c_{1}\,\frac{R\,R_{\mu\nu\rho\sigma}^{2}}{D}+c_{2}\,R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+c_{3}\,R_{\alpha\mu\nu\beta}R^{\mu\rho\sigma\nu}R_{\rho}{}^{\alpha\beta}{}_{\sigma}\right)+{\cal
O}(T^{4})\Bigg{]}\ .\end{split}$ (B.57)

The final step consists of Taylor expanding the exponential to the desired
order, namely

$\Big{\langle}e^{-S_{\rm
int}}\Big{\rangle}\Big{|}_{T^{3}}=T^{3}\left[\left(\frac{\alpha_{1}^{3}}{6}+\frac{\alpha_{1}\beta_{1}}{D}+\frac{c_{0}}{D^{2}}\right)\,\mathcal{E}_{1}+\left(\alpha_{1}\beta_{2}+\frac{c_{1}}{D}\right)\,\mathcal{E}_{2}+c_{2}(z,\omega)\,\mathcal{E}_{3}+c_{3}(z,\omega)\,\mathcal{E}_{4}\right]+\mathcal{O}(T^{4})\
.$ (B.58)

We obtained our final answer for the path integral average (2.17). In the end,
one is left only with performing the modular integration, i.e. the double
expectation value $\langle\hskip-1.42271pt\langle e^{-S_{\rm
int}}\rangle\hskip-1.42271pt\rangle$ defined in (2.16). In doing so, we can
choose the appropriate measure $P(z,\omega)$ to project only on the ghost,
graviton, or total coefficients. This gives our final results of Section 2.4,
in particular the newly-computed coefficient

$\displaystyle a_{3}(D)$
$\displaystyle=\frac{35D^{4}-147D^{3}-3670D^{2}-13560D-30240}{90720D^{2}}\;\mathcal{E}_{1}+\frac{7D^{3}-230D^{2}+3357D+12600}{15120D}\;\mathcal{E}_{2}+$
$\displaystyle\phantom{=}+\frac{17D^{2}-555D-15120}{90720}\;\mathcal{E}_{3}+\frac{D^{2}-39D-1080}{3240}\;\mathcal{E}_{4}\
.$ (B.59)

##  Appendix C Heat Kernel computations

The fourth heat kernel coefficient for the ghost field is computed from the
general formula (3.13), by performing the substitutions (3.25). It is
convenient to write

$\alpha_{3}^{\rm gh}(x)\equiv\frac{1}{7!}\bm{A}_{\rm
gh}[R,R_{\mu\nu},R_{\mu\nu\rho\sigma}]+\frac{2}{6!}\bm{B}_{\rm
gh}[R,R_{\mu\nu},R_{\mu\nu\rho\sigma},\Omega_{\mu\nu},V]\ ,$ (C.1)

where $\bm{A}_{\rm gh}$ and $\bm{B}_{\rm gh}$ are two (involved) functions of
the metric invariants and of the gauge field strength, as reported in (3.13).
Let us start from $\bm{A}_{\rm gh}$: six of the first eight terms vanish
identically since they are all proportional to covariant derivatives of $R$,
$R_{\mu\nu}$ or $R_{\mu\nu\rho\sigma}$, except from the two proportional to
$(\nabla_{\alpha}R_{\mu\nu\rho\sigma})^{2}$ and
$R_{\mu\nu\rho\sigma}\nabla^{2}R^{\mu\nu\rho\sigma}$. Recalling that
$\mbox{Tr}[\delta^{\mu}_{\nu}]=D$ and using (A.15), we are therefore left with

$\displaystyle\begin{split}\mbox{Tr}\left[\bm{A}_{\rm
gh}\right]&=D\left(3R_{\mu\nu\rho\sigma}\nabla^{2}R^{\mu\nu\rho\sigma}-\frac{208}{9}R_{\mu}{}^{\nu}R_{\nu}{}^{\sigma}R_{\sigma}{}^{\mu}+\frac{64}{3}R_{\mu\nu}R_{\rho\sigma}R^{\mu\rho\nu\sigma}-\frac{16}{3}R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\rho\sigma\tau}\right.\\\\[3.00003pt]
&\phantom{=}\left.+\frac{44}{9}R_{\mu\nu}{}^{\rho\sigma}R_{\rho\sigma}{}^{\alpha\beta}R_{\alpha\beta}{}^{\mu\nu}+\frac{80}{9}R_{\mu\nu\rho\sigma}R^{\mu\alpha\rho\beta}R^{\nu}{}_{\alpha}{}^{\sigma}{}_{\beta}\right)\\\\[3.00003pt]
&=-\frac{16}{9D}\mathcal{E}_{1}+\frac{2}{3}\mathcal{E}_{2}+\frac{17D}{9}\mathcal{E}_{3}+\frac{28D}{9}\mathcal{E}_{4}\
.\end{split}$ (C.2)

We now compute $\bm{B}_{\rm gh}$. Since
$(\Omega_{\mu\nu})^{\rho}{}_{\sigma}=R_{\mu\nu}{}^{\rho}{}_{\sigma}$, the
second term of $\bm{B}_{\rm gh}$ in (3.13) vanishes identically; moreover, as
$V\propto R$, the same occurs for the last four, leaving us with

$\displaystyle\begin{split}\mbox{Tr}\left[\bm{B}_{\rm
gh}\right]&=4\mbox{Tr}\left[\Omega_{\mu\nu}\nabla^{2}\Omega^{\mu\nu}\right]-12\mbox{Tr}\left[\Omega_{\mu}{}^{\nu}\Omega_{\nu}{}^{\sigma}\Omega_{\sigma}{}^{\mu}\right]+6\mbox{Tr}\left[R_{\mu\nu\rho\sigma}\Omega^{\mu\nu}\Omega^{\rho\sigma}\right]-4\mbox{Tr}\left[R_{\mu\nu}\Omega^{\mu\sigma}\Omega^{\nu}{}_{\sigma}\right]\\\\[3.00003pt]
&=-\frac{4}{D}\mathcal{E}_{2}-2\mathcal{E}_{3}-4\mathcal{E}_{4}\ ,\end{split}$
(C.3)

where all the traces are computed by explicit substitution, according to
(3.25),

$\displaystyle\mbox{Tr}\left[\Omega_{\mu}{}^{\nu}\Omega_{\nu}{}^{\sigma}\Omega_{\sigma}{}^{\mu}\right]$
$\displaystyle=\mbox{Tr}\left[R^{\alpha\ \ \nu}_{\ \beta\mu}R^{\beta\ \
\sigma}_{\ \gamma\nu}R^{\gamma\ \ \mu}_{\ \delta\sigma}\right]=R^{\alpha\ \
\nu}_{\ \beta\mu}R^{\beta\ \ \sigma}_{\ \gamma\nu}R^{\gamma\ \ \mu}_{\
\alpha\sigma}=-\mathcal{E}_{4}$ (C.4)
$\displaystyle\mbox{Tr}\left[R_{\mu\nu\rho\sigma}\Omega^{\mu\nu}\Omega^{\rho\sigma}\right]$
$\displaystyle=R_{\mu\nu\rho\sigma}\mbox{Tr}\left[R^{\alpha\ \mu\nu}_{\
\beta}R^{\beta\ \rho\sigma}_{\ \gamma}\right]=R_{\mu\nu\rho\sigma}R^{\alpha\
\mu\nu}_{\ \beta}R^{\beta\ \rho\sigma}_{\ \alpha}=-\mathcal{E}_{3}$ (C.5)
$\displaystyle\mbox{Tr}\left[R_{\mu\nu}\Omega^{\mu\sigma}\Omega^{\nu}{}_{\sigma}\right]$
$\displaystyle=R_{\mu\nu}\mbox{Tr}\left[R^{\alpha\ \mu\sigma}_{\
\beta}R^{\beta\ \nu}_{\ \gamma\ \sigma}\right]=R_{\mu\nu}R^{\alpha\
\mu\sigma}_{\ \beta}R^{\beta\ \nu}_{\ \alpha\
\sigma}=-\frac{1}{D}\mathcal{E}_{2}\ .$ (C.6)

Going back to (C.1) we conclude that

$\mbox{Tr}\left[\alpha_{3}^{\rm
gh}(x)\right]=-\frac{1}{2835D}\mathcal{E}_{1}+\frac{D-84}{7560D}\mathcal{E}_{2}+\frac{17D-252}{45360}\mathcal{E}_{3}+\frac{D-18}{1620}\mathcal{E}_{4}\
,$ (C.7)

which however does not provide the full coefficient for the ghost, since we
still have to add the term $\beta_{3}$ defined in (3.9), which turns out to be

$\displaystyle\begin{split}\beta_{3}^{\rm
gh}&=\frac{1}{6}\left(\alpha_{1}^{\rm
gh}\right)^{3}+\alpha_{1}^{gh}\alpha_{2}^{\rm gh}\\\
&=\frac{1}{6}\left[\delta^{\tau}_{\alpha}\left(\frac{1}{6}+\frac{1}{D}\right)R\right]^{3}+\left[\delta^{\tau}_{\alpha}\left(\frac{1}{6}+\frac{1}{D}\right)R\right]\left[\frac{1}{180}\left(R_{\mu\nu\rho\sigma}^{2}-R_{\mu\nu}^{2}\right)\delta^{\alpha}_{\gamma}+\frac{1}{12}\Omega_{\mu\nu}^{2}\right]\
,\end{split}$ (C.8)

and taking the trace

$\mbox{Tr}\left[\beta_{3}^{\rm
gh}(x)\right]=\frac{(5D^{2}+54D+180)(D+6)}{6480D^{2}}\mathcal{E}_{1}+\frac{(D+6)(D-15)}{1080D}\mathcal{E}_{2}\
.$ (C.9)

The overall result for the ghost is

$\displaystyle\begin{split}\mbox{Tr}\left[a_{3}^{\rm
gh}(x)\right]&=\frac{35D^{3}+588D^{2}+3512D+7560}{45360D^{2}}\
\mathcal{E}_{1}\\\\[5.0pt]
&\phantom{=}+\frac{7D^{2}-62D-714}{7560D}\mathcal{E}_{2}+\frac{17D-252}{45360}\mathcal{E}_{3}+\frac{D-18}{1620}\mathcal{E}_{4}\end{split}$
(C.10)

and corresponds to (2.42). To compute the fourth heat kernel coefficient for
the graviton, we start again from the general formula (3.13) and perform the
substitutions (3.26). It is once again more time convenient to split

$\alpha_{3}^{\rm gr}(x)\equiv\frac{1}{7!}\bm{A}_{\rm
gr}[R,R_{\mu\nu},R_{\mu\nu\rho\sigma}]+\frac{2}{6!}\bm{B}_{\rm
gr}[R,R_{\mu\nu},R_{\mu\nu\rho\sigma},\Omega_{\mu\nu},V]$ (C.11)

as we did in (C.1) for the ghost. By inspecting more closely the substitution
rules (3.25) and (3.26), though, it is clear that $\bm{A}_{\rm gr}$ and
$\bm{A}_{\rm gh}$ differ only by the trace of the identity operator
$\mathbbm{1}$. Using (C.2), we find in particular

$\mbox{Tr}\left[\bm{A}_{\rm
gr}\right]=\frac{D+1}{2}\,\mbox{Tr}\left[\bm{A}_{\rm
gh}\right]=(D+1)\left(-\frac{8}{9D}\mathcal{E}_{1}+\frac{1}{3}\mathcal{E}_{2}+\frac{17D}{18}\mathcal{E}_{3}+\frac{14D}{9}\mathcal{E}_{4}\right)\
,$ (C.12)

while for the computation of $\bm{B}_{\rm gr}$ we can repeat the previous
observations, and consider the only non-vanishing terms

$\displaystyle\begin{split}\mbox{Tr}\left[\bm{B}_{\rm
gr}\right]&=4\mbox{Tr}\left[\Omega_{\mu\nu}\nabla^{2}\Omega^{\mu\nu}\right]-12\mbox{Tr}\left[\Omega_{\mu}{}^{\nu}\Omega_{\nu}{}^{\sigma}\Omega_{\sigma}{}^{\mu}\right]\\\\[3.00003pt]
&\phantom{=}+6\mbox{Tr}\left[R_{\mu\nu\rho\sigma}\Omega^{\mu\nu}\Omega^{\rho\sigma}\right]-4\mbox{Tr}\left[R_{\mu\nu}\Omega^{\mu\sigma}\Omega^{\nu}{}_{\sigma}\right]+30\mbox{Tr}\left[(\nabla_{\mu}V)^{2}\right]\\\\[5.0pt]
&=-\frac{4(D+47)}{D}\ \mathcal{E}_{2}-2(D-43)\ \mathcal{E}_{3}-4(D+92)\
\mathcal{E}_{4}\end{split}$ (C.13)

where we need to compute the traces

$\displaystyle\mbox{Tr}\left[\Omega_{\mu}{}^{\nu}\Omega_{\nu}{}^{\sigma}\Omega_{\sigma}{}^{\mu}\right]$
$\displaystyle=-\left(D+2\right)\mathcal{E}_{4}$ (C.14)
$\displaystyle\mbox{Tr}\left[R_{\mu\nu\rho\sigma}\Omega^{\mu\nu}\Omega^{\rho\sigma}\right]$
$\displaystyle=-(D+2)\ \mathcal{E}_{3}$ (C.15)
$\displaystyle\mbox{Tr}\left[R_{\mu\nu}\Omega^{\mu\sigma}\Omega^{\nu}{}_{\sigma}\right]$
$\displaystyle=-\frac{D+2}{D}\ \mathcal{E}_{2}$ (C.16)
$\displaystyle\mbox{Tr}\left[\Omega_{\mu\nu}\nabla^{2}\Omega^{\mu\nu}\right]$
$\displaystyle=-(D+2)\left(\frac{2}{D}\mathcal{E}_{2}-\mathcal{E}_{3}+4\mathcal{E}_{4}\right)$
(C.17)
$\displaystyle\mbox{Tr}\left[(\nabla_{\alpha}\mathcal{V}_{\mu\nu}{}^{\rho\sigma})^{2}\right]$
$\displaystyle=-3\left(\frac{2}{D}\mathcal{E}_{2}-\mathcal{E}_{3}+4\mathcal{E}_{4}\right)\
.$ (C.18)

Going back to (C.11) we conclude that

$\displaystyle\begin{split}\mbox{Tr}\left[\alpha_{3}^{\rm
gr}(x)\right]&=-\frac{D+1}{5670D}\
\mathcal{E}_{1}+\frac{D^{2}-167D-7896}{15120D}\ \mathcal{E}_{2}\\\\[5.0pt]
&\hskip 14.22636pt+\frac{17D^{2}-487D+21672}{90720}\
\mathcal{E}_{3}+\frac{D^{2}-35D-3312}{3240}\ \mathcal{E}_{4}\ .\end{split}$
(C.19)

The last step is to compute the term $\beta_{3}$ defined in (3.9), which is
the sum of the two terms:

$\displaystyle\frac{1}{6}\left(\alpha_{1}^{\rm gr}\right)^{3}$
$\displaystyle=\frac{1}{6}\left(\frac{1}{216}R^{3}\delta_{\mu\nu}^{\ \ \
\alpha\beta}+\frac{1}{12}R^{2}\mathcal{V}_{\mu\nu}^{\ \ \
\alpha\beta}+\frac{1}{2}R\,\mathcal{V}_{\mu\nu}^{\ \ \
\rho\sigma}\mathcal{V}_{\rho\sigma}^{\ \ \
\alpha\beta}+\mathcal{V}_{\mu\nu}^{\ \ \
\rho\sigma}\mathcal{V}_{\rho\sigma}^{\ \ \
\lambda\tau}\mathcal{V}_{\lambda\tau}^{\ \ \ \alpha\beta}\right)$ (C.20)
$\displaystyle\alpha_{1}^{\rm gr}\alpha_{2}^{\rm gr}$
$\displaystyle=\left(\frac{1}{6}R\delta_{\mu\nu}^{\ \ \
\alpha\beta}+\mathcal{V}_{\mu\nu}^{\ \ \
\alpha\beta}\right)\left[\frac{1}{180}\left(R_{\mu\nu\rho\sigma}^{2}-\frac{1}{D}R^{2}\right)\delta_{\alpha\beta}^{\
\ \
\lambda\tau}+\frac{1}{12}\left(\Omega_{\rho\sigma}^{2}\right)_{\alpha\beta}^{\
\ \ \lambda\tau}\right]+\frac{1}{6}\mathcal{V}_{\mu\nu}^{\ \ \
\alpha\beta}\nabla^{2}\mathcal{V}_{\alpha\beta}^{\ \ \ \lambda\tau}$

where the following traces are to be computed:

$\displaystyle\mbox{Tr}\left[\delta_{\mu\nu}^{\ \ \ \alpha\beta}\right]$
$\displaystyle=\frac{1}{2}D(D+1)$ (C.21)
$\displaystyle\mbox{Tr}\left[\mathcal{V}_{\mu\nu}^{\ \ \ \alpha\beta}\right]$
$\displaystyle=-R$ (C.22) $\displaystyle\mbox{Tr}\left[\mathcal{V}_{\mu\nu}^{\
\ \ \rho\sigma}\mathcal{V}_{\rho\sigma}^{\ \ \ \alpha\beta}\right]$
$\displaystyle=3R^{2}_{\mu\nu\rho\sigma}$ (C.23)
$\displaystyle\mbox{Tr}\left[\mathcal{V}_{\mu\nu}^{\ \ \
\rho\sigma}\mathcal{V}_{\rho\sigma}^{\ \ \
\lambda\tau}\mathcal{V}_{\lambda\tau}^{\ \ \ \alpha\beta}\right]$
$\displaystyle=-8\mathcal{E}_{4}-\mathcal{E}_{3}$ (C.24)
$\displaystyle\mbox{Tr}\left[\mathcal{V}_{\mu\nu}^{\ \ \
\alpha\beta}\left(\Omega_{\rho\sigma}^{2}\right)_{\alpha\beta}^{\ \ \
\lambda\tau}\right]$
$\displaystyle=\frac{2}{D}\mathcal{E}_{2}+3\mathcal{E}_{3}$ (C.25)
$\displaystyle\mbox{Tr}\left[\mathcal{V}_{\mu\nu}^{\ \ \
\alpha\beta}\nabla^{2}\mathcal{V}_{\alpha\beta}^{\ \ \ \lambda\tau}\right]$
$\displaystyle=3\left(\frac{2}{D}\mathcal{E}_{2}-\mathcal{E}_{3}+4\mathcal{E}_{4}\right)$
(C.26)

leading to

$\displaystyle\begin{split}\mbox{Tr}\left[\beta_{3}^{\rm
gr}(x)\right]&=\frac{5D^{3}-D^{2}-186D+72}{12960D}\
\mathcal{E}^{3}_{1}\\\\[5.0pt]
&\phantom{=}+\frac{D^{3}-29D^{2}+468D+2520}{2160D}\
\mathcal{E}_{2}^{3}-\frac{5}{12}\mathcal{E}_{3}^{3}+\frac{2}{3}\mathcal{E}_{4}^{3}\
.\end{split}$ (C.27)

In the end,

$\displaystyle\begin{split}\mbox{Tr}\left[a_{3}^{\rm
gr}(x)\right]&=\frac{35D^{3}-7D^{2}-1318D+488}{90720D}\
\mathcal{E}_{1}+\frac{7D^{3}-202D^{2}+3109D+9744}{15120D}\
\mathcal{E}_{2}\\\\[5.0pt] &+\frac{17D^{2}-487D-16128}{90720}\
\mathcal{E}_{3}+\frac{D^{2}-35D-1152}{3240}\ \mathcal{E}_{4}\ ,\end{split}$
(C.28)

which corresponds to (2.46). By summing the ghost and graviton coefficients
according to (3.31), we end up with

$\displaystyle\mbox{Tr}\left[a_{3}(x)\right]$
$\displaystyle=\frac{35D^{4}-147D^{3}-3670D^{2}-13560D-30240}{90720D^{2}}\
\mathcal{E}_{1}+\frac{7D^{3}-230D^{2}+3357D+12600}{15120D}\ \mathcal{E}_{2}$
$\displaystyle\phantom{=}+\frac{17D^{2}-555D-15120}{90720}\
\mathcal{E}_{3}+\frac{D^{2}-39D-1080}{3240}\ \mathcal{E}_{4}\ ,$ (C.29)

which corresponds to (2.32).

##  Appendix D Topological terms in even dimensions

In order to display the cancellation of the one-loop divergences of pure
gravity in four spacetime dimensions, and analogously their resilience in six,
we exploit the Gauss-Bonnet theorem, which allows us to compute the Euler
character $\chi_{\rm E}(\mathcal{M})$ of a manifold $\mathcal{M}$ as a volume
integral of the 2-form $\mathcal{R}^{\mu\nu}\equiv R^{\mu\nu}_{\ \ \
\alpha\beta}\,\mbox{d}x^{\alpha}\wedge\mbox{d}x^{\beta}$, namely

$\chi_{\rm
E}(\mathcal{M})=\frac{1}{(4\pi)^{d}}\int_{\mathcal{M}}\varepsilon_{\mu_{1}\nu_{1}\dots\mu_{d}\nu_{d}}\mathcal{R}^{\mu_{1}\nu_{1}}\wedge\dots\wedge\mathcal{R}^{\mu_{d}\nu_{d}}\
,$ (D.1)

where $D\equiv 2d$ is the dimension of the manifold, assumed here to be
even.101010If $D$ is odd, the integral (D.1) vanishes, so that the theorem
does not provide a useful way of computing $\chi_{\rm E}(\mathcal{M})$. In
local coordinates, (D.1) becomes

$\chi_{\rm E}(\mathcal{M})=\frac{1}{2(4\pi)^{d}}\int\mbox{d}^{D}x\sqrt{g}\
\frac{D!}{D}\delta^{\alpha_{1}}_{[\mu_{1}}\delta^{\beta_{1}}_{\nu_{1}}\dots\delta^{\alpha_{d}}_{\mu_{d}}\delta^{\beta_{d}}_{\nu_{d}]}R^{\mu_{1}\nu_{1}}_{\
\ \ \ \ \alpha_{1}\beta_{1}}\dots R^{\mu_{d}\nu_{d}}_{\ \ \ \ \
\alpha_{d}\beta_{d}}\ .$ (D.2)

It is possible to prove that $\chi_{\rm E}(\mathcal{M})$ defined in this way
does not depend on the metric settled upon $\mathcal{M}$, and is fixed only by
the global topology of the manifold. For $D=2$ ($d=1$), (D.2) becomes

$\left.\chi_{\rm
E}(\mathcal{M})\right|_{D=2}=\frac{1}{16\pi}\int\mbox{d}^{2}x\sqrt{g}\ R\ ,$
(D.3)

which is proportional to Einstein-Hilbert action, while at $D=4$ ($d=2$) the
Euler character reads

$\left.\chi_{\rm
E}(\mathcal{M})\right|_{D=4}=\frac{1}{32\pi^{2}}\int\mbox{d}^{4}x\sqrt{g}\
\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\
.$ (D.4)

On Einstein spaces the first two terms in the integrand of (D.4) cancel off,
and therefore we are left with

$\left.\chi_{E}(\mathcal{M})\right|_{D=4}=\frac{1}{32\pi^{2}}\int\mbox{d}^{4}x\sqrt{g}\
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\equiv\int\mbox{d}^{4}x\sqrt{g}\
E_{4}\ .$ (D.5)

Therefore, the third heat kernel coefficient
$\mbox{Tr}\left[a_{2}(x)\right]\propto R_{\mu\nu\rho\sigma}^{2}$ is
proportional to the Euler density $E_{4}$ and hence is a total derivative,
which can be neglected in the effective action. This result is no more true
when $\Lambda\neq 0$, even if we drop the total derivative term corresponding
to Euler density.

In dimension $D=6$ ($d=3$) the Euler character is [66]

$\left.\chi_{\rm
E}(\mathcal{M})\right|_{D=6}=\frac{1}{384\pi^{3}}\int\mbox{d}^{6}x\sqrt{g}\
\left(4\mathcal{K}_{1}-48\mathcal{K}_{2}+64\mathcal{K}_{4}+96\mathcal{K}_{5}+12\mathcal{K}_{3}-96\mathcal{K}_{6}+16\mathcal{K}_{7}-32\mathcal{K}_{8}\right)\
,$ (D.6)

which on Einstein spaces reduces to

$\left.\chi_{\rm
E}(\mathcal{M})\right|_{D=6}=\frac{1}{384\pi^{3}}\int\mbox{d}^{6}x\sqrt{g}\
\left(\frac{4}{9}\mathcal{E}_{1}-4\mathcal{E}_{2}+16\mathcal{E}_{3}+32\mathcal{E}_{4}\right)\
.$ (D.7)

The condition $\Lambda=0$, that is $R=0$, forces
$\mathcal{E}_{1}=\mathcal{E}_{2}=0$, so that (D.7) eventually becomes

$\left.\chi_{\rm
E}(\mathcal{M})\right|_{D=6}=\frac{1}{384\pi^{3}}\int\mbox{d}^{6}x\sqrt{g}\
\left(16\mathcal{E}_{3}+32\mathcal{E}_{4}\right)\equiv\int\mbox{d}^{6}x\sqrt{g}\
E_{6}\ .$ (D.8)

This shows that, up to a total derivative term

$2\mathcal{E}_{4}=-\mathcal{E}_{3}\ .$ (D.9)

Thus, at dimension $D=6$, even with $\Lambda=0$, the perturbative quantum
gravity effective action is not free of logarithmic divergences.

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