# Topological Terms and the Misner String Entropy

Luca Ciambelli<EMAIL_ADDRESS>Physique Théorique et Mathématique,
Université libre de Bruxelles and International Solvay Institutes, Campus
Plaine C.P. 231, B-1050 Bruxelles, Belgium  Cristóbal Corral<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales, Facultad de Ciencias, Universidad
Arturo Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile  José
Figueroa<EMAIL_ADDRESS>Departamento de Física, Universidad de
Concepción, Casilla, 160-C, Concepción, Chile  Gastón Giribet
<EMAIL_ADDRESS>Physics Department, University of Buenos Aires and IFIBA-
CONICET, Ciudad Universitaria, Pabellón 1 (1428), Buenos Aires, Argentina
Rodrigo Olea<EMAIL_ADDRESS>Universidad Andres Bello, Departamento de
Ciencias Físicas,
Facultad de Ciencias Exactas, Sazié 2212, Piso 7, Santiago, Chile

###### Abstract

The method of topological renormalization in anti-de Sitter (AdS) gravity
consists in adding to the action a topological term which renders it finite,
defining at the same time a well-posed variational problem. Here, we use this
prescription to work out the thermodynamics of asymptotically locally anti-de
Sitter (AlAdS) spacetimes, focusing on the physical properties of the Misner
strings of both the Taub-NUT-AdS and Taub-Bolt-AdS solutions. We compute the
contribution of the Misner string to the entropy by treating on the same
footing the AdS and AlAdS sectors. As topological renormalization is found to
correctly account for the physical quantities in the parity-preserving sector
of the theory, we then investigate the holographic consequences of adding also
the Chern-Pontryagin topological invariant to the bulk action; in particular,
we discuss the emergence of the parity-odd contribution in the boundary stress
tensor.

##  I Introduction

Hawking and Hunter argued in Hawking and Hunter (1999) that the existence of
gravitational entropy is associated to a topological obstruction to foliation
of the Euclidean section of the space, which results in an obstruction for the
existence of a unitary Hamiltonian evolution. They were able to produce a
universal formula meant to express the entropy of the spacetime in terms of
defects that realize the obstructions to foliation. One example of such a
defect is given by the Misner string Misner (1963), and so this seems to imply
that the string does contribute to the entropy of the Taub-NUT spacetime Taub
(1951); Newman  _et al._ (1963). In fact, the prime examples chosen in Hawking
and Hunter (1999) to illustrate the phenomenon were precisely spaces with non-
vanishing NUT charge, for which it was found that the entropy was not just a
quarter the area, as it is for usual black holes Bekenstein (1973). This
feature was further studied in Hawking  _et al._ (1999), where asymptotically
locally anti-de Sitter (AlAdS) spacetimes were considered. For both Taub-Nut-
AdS and Taub-Bolt-AdS solutions it was found that the additional contribution
to the entropy coming from the Misner strings was essential to recover the
entropy obtained from the usual Euclidean action calculation. This result for
the entropy, on the other hand, seemed consistent with the AdS/CFT
expectations. Indeed, soon later, Emparan, Johnson, and Myers reexamined the
problem in AdS using holographic renormalization techniques Emparan  _et al._
(1999). The addition of boundary counterterms to the gravitational action
renders the Euclidean action computation of AlAdS sacetimes finite and
independent of the background subtraction. With this holography inspired
method, finite quantities for Taub-NUT/Bolt-AdS spacetimes can be derived.

The question as to whether gravitational entropy can be ascribed to spacetimes
containing defects such as Misner strings, independently of the existence of
event horizons, was also discussed in Mann (1999, 2000), again using
holographic renormalization techniques. By evaluating the entropy and Noether
charges of Kerr-NUT/Bolt-AdS spacetimes, it was concluded in Mann (2000) that,
whenever the NUT charge is non-zero, the entropy does not equal one quarter of
the area due to the contribution of the Misner string, in agreement with the
observations of Hawking and Hunter (1999); Hawking  _et al._ (1999).

In these twenty years, the geometry and thermodynamics of Taub-NUT/Bolt
spacetimes have been studied by many authors and in many different contexts
Garfinkle and Mann (2000); Clarkson  _et al._ (2003a, b); Astefanesei  _et
al._ (2005); Padmanabhan (2011); Johnson (2014); Clément  _et al._ (2015);
Araneda  _et al._ (2016); Hennigar  _et al._ (2019); Ballon Bordo  _et al._
(2019a, b); Donnay and Giribet (2019); Durka (2019); Kol and Porrati (2020);
Porrati (2020); Kalamakis  _et al._ (2020). Recently, special attention has
been paid to the presence or absence of the Misner string Johnson (2014);
Hennigar  _et al._ (2019); Ballon Bordo  _et al._ (2019a, b). Here, we analyze
the problem by using the method of topological renormalization; meaning, the
method that consists of adding to the gravitational action a bulk piece of the
Chern-Weil-Gauss-Bonnet (hereafter, Gauss-Bonnet) topological invariant, with
the specific value of the coupling constant that renders the action equivalent
to the MacDowell-Mansouri one MacDowell and Mansouri (1977); Townsend (1977).
This procedure yields finite results for the Euclidean action, the Noether
charges, and the thermodynamic quantities, while at the same time suffices to
render the variational principle well-posed Aros  _et al._ (2000a, b); see
also Olea (2005, 2007); Miskovic and Olea (2009); Giribet  _et al._ (2018).
The topological renormalization method has been shown to be consistent with
holographic renormalization Miskovic and Olea (2009); Anastasiou  _et al._
(2020), and so it will permit to analyze the problem in a way that is
independent of the background substraction. In fact, we will show below that,
while treating the AdS and AlAdS sectors in equal footing and in a unified
scheme, we will obtain results for the thermodynamics of Taub-NUT/Bolt-AdS
spacetimes that are in full agreement with the results found in the
literature. In particular, we will analyze in detail the contribution to the
entropy computation that comes from the Misner string.

The use of topological invariants to support the original Hawking-Hunter
argument makes more explicit the link between gravitational entropy and
topological obstructions. It is thus natural to explore the effects of
introducing other topological invariants and the consequences in the context
of holography. The other topological invariant consisting in a dimension-four
operator in four dimensions is the Chern-Pontryagin invariant, which carries
opposite (odd) parity with respect to the Gauss-Bonnet one. The Chern-
Pontryagin term does not affect the renormalization of the gravitational
action, so its coupling constant is a priori arbitrary. The latter is
reminiscent of the $\theta$-angle in gauge theories and can similarly be fixed
by minimizing the value of the action when evaluated on self-dual solutions.
We show that, adding this invariant to the bulk action, the boundary
holographic stress tensor is augmented by the Hodge-dual Cotton tensor of the
boundary. The full boundary stress tensor has now both even and odd parity
contributions which realizes the so-called stress tensor/Cotton tensor duality
Bakas (2009); Mukhopadhyay  _et al._ (2014). Combining the Cotton and stress
tensor together has been instrumental in the fluid/gravity correspondence
Leigh  _et al._ (2012a, b); Caldarelli  _et al._ (2011); Petropoulos and
Siampos (2015); Ciambelli  _et al._ (2017). This comes about thanks to the
observation that the bulk Weyl tensor asymptotes to a specific combination of
the latter Leigh and Petkou (2007); Mansi  _et al._ (2009a, b). Since this
combination is complex in Lorentzian signature, its parity odd piece, whose
physical interpretation being challenging, has been dubbed “reference tensor”;
still, its contribution has shown to be important in the fluid/gravity fill-in
problem Gath  _et al._ (2015), where it controls the Petrov class in the bulk.
Here, on top of boundary considerations, we explore the consequences of adding
the bulk ancestor of the Cotton tensor –the Chern-Pontryagin term– on the
entropy.

The paper is organized as follows. In Sec. II, we review the fundamentals of
Wald’s formalism, which yields a procedure to compute the gravitational
entropy. Section III is the bulk of the manuscript: we start showing the
consequences on the Noether charge of adding the Gauss-Bonnet term to the bulk
action, then we introduce the Taub-NUT/Bolt-AdS solutions and prove that the
renormalized entropy and thermodynamics is recovered when the Gauss-Bonnet
coupling constant is fixed to cancel divergences. Finally we explore the
consequences of the Chern-Pontryagin term both in the bulk thermodynamics and
the boundary theory. We conclude offering some final comments and possible
outlooks in Sec. IV.

##  II Noether-Wald formalism

In this section, we review the basics of Noether-Wald’s formalism Wald (1993);
Iyer and Wald (1994) for the sake of completeness and to state conventions.
Let us start by considering a general gravitational action principle defined
on a $D$-dimensional manifold $\mathcal{M}$ provided with a metric
$g_{\mu\nu}$ as

$\displaystyle
I\left[g_{\mu\nu}\right]=\int_{\mathcal{M}}\text{d}^{D}x\sqrt{-g}\;\mathscr{L}\left[g_{\mu\nu},R^{\lambda\rho}{}_{\mu\nu}\right],$
(1)

where $R^{\lambda\rho}{}_{\mu\nu}=g^{\rho\sigma}R^{\lambda}{}_{\sigma\mu\nu}$
is the Riemann tensor and $g=\det g_{\mu\nu}$. The metric is considered here
as the only dynamical field: the connection is assumed to be torsion-free and
metric compatible.111For extensions where these assumptions are relaxed see
Nester (1991); Chen and Nester (1999).

A stationary variation of the action (1) yields Wald (1993); Iyer and Wald
(1994); Wald and Zoupas (2000); Padmanabhan (2011)

$\displaystyle\delta I$
$\displaystyle=\delta\int_{\mathcal{M}}\text{d}^{D}x\sqrt{-g}\,\mathscr{L}=\int_{\mathcal{M}}\text{d}^{D}x\sqrt{-g}\,\delta
g^{\mu\nu}\mathcal{E}_{\mu\nu}+\int_{\mathcal{M}}\text{d}^{D}x\sqrt{-g}\,\nabla_{\mu}\Theta^{\mu},$
(2)

where

$\displaystyle\mathcal{E}_{\mu\nu}$
$\displaystyle=E_{\mu}{}^{\lambda\rho\sigma}R_{\nu\lambda\rho\sigma}-\frac{1}{2}g_{\mu\nu}\mathscr{L}-2\nabla^{\lambda}\nabla^{\rho}E_{\mu\lambda\rho\nu},$
(3) $\displaystyle\Theta^{\mu}$
$\displaystyle=2\delta\Gamma^{\lambda}{}_{\nu\rho}E_{\lambda}{}^{\rho\mu\nu}-2\delta
g_{\nu\sigma}\nabla_{\rho}E^{\nu\mu\rho\sigma}=-2\nabla_{\rho}\delta
g_{\nu\sigma}E^{\rho\sigma\mu\nu}+2\delta
g_{\nu\sigma}\nabla_{\rho}E^{\rho\sigma\mu\nu}.$ (4)

Here, $\mathcal{E}_{\mu\nu}=0$ denotes the field equations and we have defined
the functional derivative of the Lagrangian with respect to the Riemannian
curvature as222From hereon, we assume that
$E_{\mu\nu\lambda\rho}=-E_{\nu\mu\lambda\rho}$,
$E_{\mu\nu\lambda\rho}=-E_{\mu\nu\rho\lambda}$,
$E_{\mu\nu\lambda\rho}=E_{\lambda\rho\mu\nu}$, and
$E_{\mu[\nu\lambda\rho]}=0$.

$\displaystyle
E_{\lambda\rho}{}^{\mu\nu}\equiv\frac{\partial\mathscr{L}}{\partial
R^{\lambda\rho}{}_{\mu\nu}}.$ (5)

This tensor plays a crucial role in Noether-Wald’s formalism. The second term
in Eq. (2) is the boundary term arising from the variation of the action and
it generically depends on the fields and variations thereof. By means of the
Stokes’ theorem, it can be expressed as

$\displaystyle\int_{\mathcal{M}}\text{d}^{D}x\sqrt{-g}\;\nabla_{\mu}\Theta^{\mu}$
$\displaystyle=\int_{\partial\mathcal{M}}\text{d}^{D-1}x\sqrt{h}\;n_{\mu}\Theta^{\mu}$
$\displaystyle=-2\int_{\partial\mathcal{M}}\text{d}^{D-1}x\sqrt{h}\;n_{\mu}\left[\nabla_{\rho}\delta
g_{\nu\sigma}E^{\rho\sigma\mu\nu}-\delta
g_{\nu\sigma}\nabla_{\rho}E^{\rho\sigma\mu\nu}\right],$ (6)

where $h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$ is the induced metric on the
boundary, $h$ its determinant, and $n_{\mu}$ is a space-like unit normal such
that $n_{\mu}n^{\mu}=1$ with $n^{\mu}h_{\mu\nu}=0$.

Equation (2) implies that diffeomorphism invariance leads to

$\displaystyle\nabla_{\mu}\left[\Theta^{\mu}(g,\mathcal{L}_{\xi}g)-\xi^{\mu}\mathscr{L}\right]\equiv\nabla_{\mu}J^{\mu}=-\mathcal{L}_{\xi}g^{\mu\nu}\mathcal{E}_{\mu\nu},$
(7)

where $\mathcal{L}_{\xi}$ is the Lie derivative along the vector field $\xi$
and

$\displaystyle
J^{\mu}=-2\nabla_{\nu}\left(E^{\mu\nu\rho\sigma}\nabla_{\rho}\xi_{\sigma}+2\xi_{\rho}\nabla_{\sigma}E^{\mu\nu\rho\sigma}\right).$
(8)

The Noether current $J^{\mu}$ is conserved on-shell as it can be read off from
Eq. (7). The Poincaré lemma, in turn, implies that locally the Noether current
$J^{\mu}$ can be written as

$\displaystyle
J^{\mu}=\nabla_{\nu}q^{\mu\nu}\;\;\;\;\;\mbox{where}\;\;\;\;\;q^{\mu\nu}=-2\left(E^{\mu\nu\rho\sigma}\nabla_{\rho}\xi_{\sigma}+2\xi_{\rho}\nabla_{\sigma}E^{\mu\nu\rho\sigma}\right)=-q^{\nu\mu},$
(9)

is known as the Noether prepotential. Thus, the conserved Noether charge
associated to diffeomorphism invariance generated by the vector field $\xi$ is

$\displaystyle
Q[\xi]=\frac{1}{2}\int_{\Sigma}\epsilon_{\mu_{1}\ldots\mu_{D}}q^{\mu_{1}\mu_{2}}\text{d}x^{\mu_{3}}\wedge\ldots\wedge\text{d}x^{\mu_{D}}\equiv\int_{\Sigma}Q_{\mu_{1}\ldots\mu_{D-2}}\text{d}x^{\mu_{1}}\wedge\ldots\wedge\text{d}x^{\mu_{D-2}}=\int_{\Sigma}\mathbf{Q},$
(10)

where $\Sigma$ is a codimension-2 hypersurface. According to Wald (1993), the
entropy is obtained when $\xi$ is an asymptotically time-like Killing vector
and $\Sigma$ corresponds to the bifurcating horizon. In presence of additional
obstructions to foliation, e.g. topological defects, other contributions to
the entropy arise Garfinkle and Mann (2000). In general, it can be expressed
as

$\displaystyle S=\beta_{\tau}\int_{\Sigma}\mathbf{Q},$ (11)

where $\beta_{\tau}$ is the period of the Euclidean time for the avoidance of
conical singularities.

In the case of black holes in Einstein theory, Eq. (11) implies that the
entropy is one quarter of the horizon’s area. However, a remarkable
counterexample appears in Taub-NUT/Bolt geometry with spherical base manifold,
since the presence of Misner strings breaks down the standard entropy/area
relation Hawking  _et al._ (1999); Emparan  _et al._ (1999); Mann (1999);
Garfinkle and Mann (2000); Astefanesei  _et al._ (2005). The modification of
the entropy law stems from the obstruction to foliate the spacetime with
constant-time hypersurfaces, turning the contribution from the Misner string
nontrivial at the poles. Through the Noether-Wald formalism, this was computed
in Ref. Garfinkle and Mann (2000) for asymptotically locally flat (AlF)
spaces. In AdS, in contrast, the entropy of the Misner string defined by
Wald’s formulae becomes divergent. In the following, we show that this problem
can be circumvented by introducing the Gauss-Bonnet invariant into the
gravitational action.

##  III Topological renormalization

In four dimensions, the Einstein-Hilbert action in presence of the Gauss-
Bonnet term is333We call this the Einstein-Gauss-Bonnet action and its
integrand the Einstein-Gauss-Bonnet Lagrangian.

$\displaystyle I_{\rm
EGB}[g_{\mu\nu}]=\kappa\int\text{d}^{4}x\sqrt{-g}\left(R-2\Lambda+\alpha\mathcal{G}\right),$
(12)

where $\kappa=\left(16\pi G\right)^{-1}$ is the gravitational constant,
$\alpha$ is the Gauss-Bonnet coupling constant and

$\displaystyle\mathcal{G}$
$\displaystyle=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}=3!\delta_{[\mu}^{[\alpha}\delta_{\nu}^{\beta}\delta_{\lambda}^{\gamma}\delta_{\rho]}^{\delta]}R^{\mu\nu}{}_{\alpha\beta}R^{\lambda\rho}{}_{\gamma\delta}.$
(13)

The field equations obtained from the action (12) and the off-shell functional
derivative of the Einstein-Gauss-Bonnet Lagrangian with respect to the Riemann
tensor are

$\displaystyle\mathcal{E}_{\mu\nu}$
$\displaystyle=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=0,$ (14)
$\displaystyle E^{\mu\nu}{}_{\lambda\rho}$
$\displaystyle=\kappa\left(\delta_{[\lambda}^{[\mu}\delta_{\rho]}^{\nu]}+12\alpha\delta_{[\lambda}^{[\mu}\delta_{\rho}^{\nu}\delta_{\alpha}^{\gamma}\delta_{\beta]}^{\delta]}R^{\alpha\beta}{}_{\gamma\delta}\right),$
(15)

respectively. Equation (14) is solved by Einstein spaces with
$R_{\mu\nu}=\Lambda g_{\mu\nu}$. Thus, on-shell, the Riemann tensor can be
written as

$\displaystyle R^{\mu\nu}{}_{\lambda\rho}$
$\displaystyle=W^{\mu\nu}{}_{\lambda\rho}-\frac{2}{\ell^{2}}\delta_{[\lambda}^{[\mu}\delta_{\rho]}^{\nu]},$
(16)

where $\Lambda=-3/\ell^{2}$ and the Weyl tensor has been defined as

$W_{\mu\nu\lambda\rho}=R_{\mu\nu\lambda\rho}+\frac{1}{2}\Big{(}R_{\mu\rho}g_{\nu\lambda}-R_{\nu\rho}g_{\mu\lambda}+R_{\nu\lambda}g_{\mu\rho}-R_{\mu\lambda}g_{\nu\rho}\Big{)}+\frac{1}{6}R\Big{(}g_{\mu\lambda}g_{\nu\rho}-g_{\mu\rho}g_{\nu\lambda}\Big{)}.$
(17)

For Einstein-AdS spaces, Eq. (15) becomes

$\displaystyle E^{\mu\nu}{}_{\lambda\rho}$
$\displaystyle=\kappa\left(1-\frac{4\alpha}{\ell^{2}}\right)\delta_{[\lambda}^{[\mu}\delta_{\rho]}^{\nu]}+2\kappa\alpha
W^{\mu\nu}{}_{\lambda\rho},$ (18)

and, therefore, the Noether prepotential reads

$\displaystyle
q^{\mu\nu}=-2\kappa\left[\left(1-\frac{4\alpha}{\ell^{2}}\right)\delta_{[\lambda}^{[\mu}\delta_{\rho]}^{\nu]}+2\alpha
W^{\mu\nu}{}_{\lambda\rho}\right]\nabla^{\lambda}\xi^{\rho}.$ (19)

It is well known that there exists a proper choice of the Gauss-Bonnet
coupling in four dimensions renormalizing the Noether charge Aros  _et al._
(2000a) and Euclidean on-shell action Olea (2005) for AlAdS solutions. Its
particular value is obtained by demanding that the first term on the right-
hand side of Eq. (18) vanishes, which is justified by the fact the Weyl tensor
is the only combination between the curvature and the metric that has the
correct falloff.444The appearance of the Weyl tensor at the boundary is the
key ingredient to make contact with the notion of Conformal Mass in AAdS
spacetimes Ashtekar and Magnon (1984); Jatkar  _et al._ (2014).

The very same choice allows one to renormalize the entropy of Misner string in
Euclidean Taub-NUT/Bolt-AdS solutions. This is remarkable as one could have
expected that the coefficient that suffices to renormalize solutions in the
asymptotically AdS sector would differ from the one that does the job in other
AlAdS sectors. More concretely, since different values of the NUT charge
define different asymptotic sectors, one could have expected that the coupling
of the topological term would in general depend on that charge, but notably it
does not. Here, we will show that a similar phenomenon occurs in the
thermodynamics calculation, where the integration is not performed at
infinity.

###  III.1 Taub-NUT/Bolt-AdS

The field equations (14) are solved by the Euclidean inhomogeneous stationary
metric constructed on complex line bundles over $S^{2}$, that is

$\displaystyle\text{d}s^{2}$
$\displaystyle=f(r)\left(\text{d}\tau+2n\cos\theta\,\text{d}\phi\right)^{2}+\frac{\text{d}r^{2}}{f(r)}+(r^{2}-n^{2})\left(\text{d}\theta^{2}+\sin^{2}\theta\,\text{d}\phi^{2}\right),$
(20)

where

$\displaystyle
f(r)=\frac{r^{2}+n^{2}}{r^{2}-n^{2}}-\frac{2MGr}{r^{2}-n^{2}}+\frac{r^{4}-6r^{2}n^{2}-3n^{4}}{\ell^{2}\left(r^{2}-n^{2}\right)},$
(21)

and $\Lambda=-3/\ell^{2}$. This is the asymptotically AdS version Hawking  _et
al._ (1999) of the Taub-NUT spacetime Taub (1951); Newman  _et al._ (1963),
whose Euclidean version can be regarded as a gravitational instanton Eguchi
and Hanson (1978); Gibbons and Hawking (1978). Here, $M$ is an integration
constant associated to the mass, while $n$ is a constant that can be thought
of as the gravitational magnetic charge $N=n/G$. Nuts and bolts, alongside the
absence of conical singularities, are characterized by

For NUT: $\displaystyle f(n)$ $\displaystyle=0$ and $\displaystyle
f^{\prime}(r)\big{|}_{r=n}$ $\displaystyle=\frac{4\pi}{\beta_{\tau}},$ (22)
For Bolt: $\displaystyle f(r_{b})$ $\displaystyle=0$ and $\displaystyle
f^{\prime}(r)\big{|}_{r=r_{b}}$ $\displaystyle=\frac{4\pi}{\beta_{\tau}},$
(23)

where $r_{b}>n$ is the bolt radius. These conditions determine the set of
fixed points to be one- and two-dimensional, respectively. In turn, they fix
the integration constant $M$ as

$\displaystyle M$
$\displaystyle=\frac{n}{G}\left(1-\frac{4n^{2}}{\ell^{2}}\right)\equiv M_{\rm
NUT},$ (24) $\displaystyle M$
$\displaystyle=\frac{1}{2Gr_{b}}\left(r_{b}^{2}+n^{2}+\frac{r_{b}^{4}-6r_{b}^{2}n^{2}-3n^{4}}{\ell^{2}}\right)\equiv
M_{\rm Bolt},$ (25)

respectively. For NUT, the period of the Euclidean time is $\beta_{\tau}=8\pi
n$, and the Weyl tensor is globally self-dual. That implies that the total
mass, in terms of the electric and magnetic mass of the solution is
identically zero Araneda  _et al._ (2016). Therefore, for different values of
the NUT charge, this solution represents topologically inequivalent vacuum
states. For bolt, on the other hand, the Weyl tensor is not globally self-dual
and the period of the Euclidean time is given by

$\displaystyle\beta_{\tau}$ $\displaystyle=\frac{4\pi
r_{b}}{1+\frac{3}{\ell^{2}}\left(r_{b}^{2}-n^{2}\right)}.$ (26)

Additionally, the requirement of the Misner string being unobservable555By
unobservable we mean that bolt solutions should have the same temperature as
NUT ones. As we will see, they still have different entropy though. Misner
(1963) imposes a further condition, i.e. $\beta_{\tau}=8\pi n$, which relates
the bolt radius with the NUT charge according to

$\displaystyle r_{b}$
$\displaystyle=\frac{\ell^{2}}{12n}\left[1\pm\sqrt{1-\frac{48n^{2}}{\ell^{2}}\left(1-\frac{3n^{2}}{\ell^{2}}\right)}\right].$
(27)

Reality and positivity of the bolt radius impose a range on the NUT charge
such that the solution exists.

The role of the Gauss-Bonnet term is crucial to renormalize the entropy
contribution of the Misner string. The relevant components of the Noether
charge (10) are computed by inserting the off-shell value of (15) into Eq.
(9); they read

$\displaystyle Q_{\theta\phi}$
$\displaystyle=\kappa\left(r^{2}-n^{2}\right)f^{\prime}\sin\theta$
$\displaystyle+\frac{4\kappa\alpha}{\left(r^{2}-n^{2}\right)^{2}}\left[\left(r^{2}-n^{2}\right)^{2}f^{\prime}-\left(r^{4}+4r^{2}n^{2}-5n^{4}\right)f^{\prime}f+4f^{2}rn^{2}\right]\sin\theta,$
(28) $\displaystyle Q_{r\phi}$ $\displaystyle=-\frac{4\kappa
n^{2}f\cos\theta}{r^{2}-n^{2}}+\frac{8\kappa\alpha
n^{2}}{\left(r^{2}-n^{2}\right)^{2}}\left[\left(r^{2}-n^{2}\right)\left(f^{\prime\prime}f+f^{\prime
2}\right)-2f^{\prime}fr\right]\cos\theta.$ (29)

Thus, the Noether charge in this case is

$\displaystyle\int_{\Sigma}\mathbf{Q}$
$\displaystyle=\int_{r=r_{b}}\mathbf{Q}+\int_{\theta=0}\mathbf{Q}+\int_{\theta=\pi}\mathbf{Q}$
$\displaystyle=\int_{0}^{2\pi}\text{d}\phi\int_{0}^{\pi}\text{d}\theta\;Q_{\theta\phi}\big{|}_{r=r_{b}}-\int_{0}^{2\pi}\text{d}\phi\int_{r_{b}}^{\infty}\text{d}r\;Q_{r\phi}\big{|}_{\theta=0}+\int_{0}^{2\pi}\text{d}\phi\int_{r_{b}}^{\infty}\text{d}r\;Q_{r\phi}\big{|}_{\theta=\pi}\,,$
(30)

where the last two terms arise from the contribution of the Misner string. In
absence of the cosmological constant, these integrals were computed in Ref.
Garfinkle and Mann (2000) giving $S=4\pi n^{2}$ and $S=5\pi n^{2}$ for nuts
and bolts, respectively. Their difference yields to the same result as in Ref.
Hawking  _et al._ (1999) without infinite background subtraction. In presence
of the cosmological constant, however, the last two terms are divergent. One
possibility to circumvent this problem is to add proper counterterms and
employ Euclidean methods as in Ref. Mann (1999). The same procedure leads to
the aforementioned entropies in absence of the cosmological constant.

Here, we notice that an alternative method to renormalize the contribution of
the Misner string in AdS is to fix the Gauss-Bonnet coupling as

$\displaystyle\alpha=\frac{\ell^{2}}{4}.$ (31)

With this choice, the action (12) becomes the one of MacDowell-Mansouri666In
five dimensions the quadratic Gauss-Bonnet term does change the theory at
classical level, and for the same choice of couplings constants (31) the
action of the gravitational theory coincides with that of five-dimensional
Chern-Simons; see Zanelli (2005) and references therein. MacDowell and
Mansouri (1977); Townsend (1977) and, on-shell, it can be written as the
conformal invariant contribution Araneda  _et al._ (2016)

$I_{\text{EGB}}=\frac{\kappa\ell^{2}}{4}\int\text{d}^{4}x\sqrt{-g}\,W_{\mu\nu\lambda\rho}W^{\mu\nu\lambda\rho}.$
(32)

Additionally, the Noether prepotential for Einstein spaces becomes

$\displaystyle
q^{\mu\nu}=-\kappa\ell^{2}\,W^{\mu\nu}{}_{\lambda\rho}\nabla^{\lambda}\xi^{\rho}.$
(33)

Thus, Eq. (11) yields the renormalized entropy

$\displaystyle S_{\rm Bolt}$ $\displaystyle=\frac{32\pi^{2}\kappa
n}{r_{b}}\left[4r_{b}^{2}-2n^{2}+\frac{1}{\ell^{2}}\left(3r_{b}^{4}-12r_{b}^{2}n^{2}-3n^{4}+\ell^{4}\right)\right],$
(34)

which, after replacing (27), can be seen to agree with the result in the
literature Johnson (2014) up to a (thermodynamic irrelevant) constant piece
$\Delta S=\pi\ell^{2}/G$. It is worth noticing that we could have done the
computation in a different gauge by considering in (20) the Misner change of
coordinate $\tau\to\tau\pm 2n\phi$, which suffices to eliminate one of the
Misner strings, namely the one located at $\theta=\pi/2(1\pm 1)$, at the price
of introducing closed timelike curves in the Lorentzian geometry. As expected,
the result of the computation performed in this way agrees with (34), with one
of the last terms in the second line of (30) contributing zero in that case,
but being it compensated by the other.

The advantage of using Noether-Wald procedure for the $4D$ Einstein-Gauss-
Bonnet action is that the resulting formula is fully covariant. Thus, it
properly adapts to include all the boundaries present in the geometry. In
addition, it can systematically be generalized to higher, even dimensions.

The lowest order in the saddle-point approximation from the quantum
statistical relation $\ln Z\approx-I$, where $Z$ is the partition function and
$I$ is the on-shell Euclidean action, provides a correct thermodynamic
description of gravitational solutions. Remarkably, the same result for the
entropy can be obtained by choosing the Gauss-Bonnet coupling in Eq. (31),
such that the renormalized Euclidean action is

$\displaystyle I_{\rm Bolt}=-\frac{32\pi^{2}\kappa
n}{r_{b}}\left[2r_{b}^{2}-4n^{2}+\frac{1}{\ell^{2}}\left(r_{b}^{4}+3n^{4}+\ell^{4}\right)\right].$
(35)

It is straightforward to check that this expression yields to the same entropy
obtained from the Wald’s formalism [cf. Eq. (34)] through

$\displaystyle S_{\rm Bolt}=\beta_{\tau}\frac{\partial I_{\rm
Bolt}}{\partial\beta_{\tau}}-I_{\rm Bolt}.$ (36)

Thus, the Gibbs-Duhem relation $S=\beta_{\tau}H_{\infty}-I$ allows one to
obtain the energy of the system as

$\displaystyle H_{\infty}$
$\displaystyle=\frac{8\pi\kappa}{r_{b}}\left[r_{b}^{2}+n^{2}+\frac{1}{\ell^{2}}\left(r_{b}^{4}-6r_{b}^{2}n^{2}-3n^{4}\right)\right],$
(37)

which is exactly the same as $M$ in Eq. (25).

Notice that, when comparing with the results in Johnson (2014), the entropy
(34) has an additional contribution $\Delta S=\pi\ell^{2}/G$. In the case of
topological black holes, this shift in the entropy is proportional to the
Euler characteristic of the horizon Olea (2005); i.e. it computes a
topological number of the constant-$\tau$ slices of the horizon. As $\Delta S$
is just an additive constant, it does not affect relevant thermodynamic
relations such as the first law. On the other hand, the value of (35) differs
from those in Johnson (2014); Emparan  _et al._ (1999) by a term $\Delta
I=-\Delta S$ so that (36) holds, showing that $\Delta S$ indeed reflects the
horizon topology. For $r_{b}=n$, the result (34) takes the form

$\displaystyle S_{\rm NUT}$ $\displaystyle=\frac{4\pi
n^{2}}{G}\left(1-\frac{6n^{2}}{\ell^{2}}\right)+\frac{2\pi\ell^{2}}{G},$ (38)

which also agrees with the entropy of the Taub-NUT-AdS solution given in Mann
(1999); Emparan  _et al._ (1999); Johnson (2014) up to a constant, cf. Hawking
_et al._ (1999).777Our notation relates to that of Hawking  _et al._ (1999) by
identifying parameters as $\ell=2b,n=b\sqrt{E},k=1,s=r_{b}/(b\sqrt{E})$ and
rescaling the time and radial coordinates by $2b$ and $b\sqrt{E}$,
respectively.

###  III.2 Adding the Chern-Pontryagin invariant

Having shown above that the addition of the Gauss-Bonnet invariant suffices to
renormalize the gravitational action and yields the correct result for the
charges, it is natural to ask about the effects of including the only other
dimension-four topological operator in four dimensions, i.e. the Chern-
Pontryagin topological invariant. The latter reads

${\cal P}_{4}=-\frac{1}{2}\star R^{\mu\nu}_{\ \rho\eta}R_{\
\mu\nu}^{\rho\eta}=-\frac{1}{4}\varepsilon^{\mu\nu}_{\
\alpha\beta}R^{\alpha\beta}_{\ \rho\eta}R^{\rho\eta}_{\ \mu\nu},$ (39)

where we defined the left Hodge dual of the Riemann tensor following the
convention $\star
R_{\mu\nu\rho\eta}\equiv\tfrac{1}{2}\varepsilon_{\mu\nu\alpha\beta}R^{\alpha\beta}{}_{\rho\eta}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}R^{\alpha\beta}_{\
\rho\eta}$. Evaluated on the Taub-NUT-AdS geometry this yields

$-\frac{1}{8\pi^{2}}\int d^{4}x\,\sqrt{-g}\,{\cal
P}_{4}=\frac{1}{32\pi^{2}}\int d^{4}x\,\sqrt{-g}\,\varepsilon^{\mu\nu}_{\
\alpha\beta}R^{\alpha\beta}_{\ \rho\eta}R^{\rho\eta}_{\
\mu\nu}=2-\frac{16n^{2}}{\ell^{2}}\left(1-\frac{2n^{2}}{\ell^{2}}\right)\,.$
(40)

This value, which reduces to the well-known result $2$ for the Ricci flat
metric in the limit $\ell\to\infty$, changes its global sign if one perform
the change $n\to-n$ in (20) (see Ref. Hawking (1977)). We also see from this
expression that something special occurs for $n=\pm\ell/2$, for which (40)
vanishes. Notice that the mass (24) is also zero for such especial values of
$n$. As discussed in Johnson (2014), the geometries with $n=\pm\ell/2$
correspond to AdS4 spacetime with a non-trivial slicing, the trivial
$S^{2}\times S^{1}$ slicing corresponding to $n=M=0$. It is worth noticing
that for $n=\pm\ell/2$ the entropy formula (38) gives a positive result,
avoiding the puzzle of negative entropy observed in Chamblin  _et al._ (1999);
Mann (1999); Johnson (2014). Notice also that for $n=\pm\ell/\sqrt{2}$ the
value of the signature (40) reduces to that of Ricci flat Taub-NUT. It is the
dimensionless ratio $n/\ell$ what controls the value of the Pontryagin
invariant, so suggesting that the values that this ratio takes might be
important for the quantum theory.

The Chern-Pontryagin term (39) has opposite parity than the Gauss-Bonnet one,
such that its inclusion results in an action that is not longer parity even.
As the ${\cal P}_{4}$ can be written as

${\cal P}_{4}=-\frac{1}{2}\star W^{\mu\nu}_{\ \rho\eta}W^{\rho\eta}_{\
\mu\nu},$ (41)

when adding it to the gravitational Lagrangian one finds the on-shell
renormalized action

$I_{\text{EGBP}}[g_{\mu\nu}]=\frac{\kappa\ell^{2}}{4}\int
d^{4}x\sqrt{-g}\,W^{\mu\nu}_{\ \rho\eta}W^{\rho\eta}_{\
\mu\nu}+\vartheta\,\frac{\kappa\ell^{2}}{4}\int d^{4}x\sqrt{-g}\,\star
W^{\mu\nu}_{\ \rho\eta}W^{\rho\eta}_{\ \mu\nu},$ (42)

where $\vartheta$ is a constant that is reminiscent of the coupling of the
$\theta$-term in gauge theory. In fact, the action (42) can be thought of as
the MacDowell-Mansouri action augmented with a $CP$ violating term. In
contrast to what happens with the coupling $\alpha$ of the Gauss-Bonnet
invariant, which can be fixed as in (31) by demanding regularity of the
action, the value of $\vartheta$ is not fixed by such a requirement.
Therefore, it is necessary to apply a different criterion. A natural one would
be asking the action to vanish when evaluated on the self-dual vacuum
solution. This results in the value888The two signs here correspond to the
self-dual and the anti-self-dual solutions.

$\vartheta=\pm 1\,.$ (43)

Using this, the on-shell action can be written as Araneda  _et al._ (2016)

$I_{\text{EGBP}}=\frac{\kappa\ell^{2}}{32}\int_{\mathcal{M}}\sqrt{-g}\;\delta_{[\mu}^{[\alpha}\delta_{\nu}^{\beta}\delta_{\lambda}^{\sigma}\delta_{\rho]}^{\tau]}\left(W^{\mu\nu}{}_{\alpha\beta}\pm\star
W^{\mu\nu}{}_{\alpha\beta}\right)\left(W^{\lambda\rho}{}_{\sigma\tau}\pm\star
W^{\lambda\rho}{}_{\sigma\tau}\right).$ (44)

Since the Taub-NUT-AdS solution with $M$ fixed by Eq. (24) is globally
(anti-)self dual, the action (44) vanishes identically when evaluated at this
configuration. For bolt, we take the case $n>0$ for the sake of simplicity.
Thus, the renormalized on-shell action in the presence of the Chern-Pontryagin
term with fixed theta parameter is

$\displaystyle I_{\rm Bolt}=-\frac{16\pi^{2}\kappa
n\left(2r_{b}+n\right)}{r_{b}^{2}}\left[2\left(r_{b}-n\right)\left(r_{b}+3n\right)+\frac{1}{\ell^{2}}\left[\left(r_{b}-n\right)^{2}\left(r_{b}+3n\right)^{2}+\ell^{4}\right]\right],$
(45)

which, in contrast to (35), for $r_{b}=n$ is a constant independent on $n$.
This is expected: since the action (44) vanishes when evaluated at the Taub-
NUT-AdS solution, Eq. (45) can differ from zero only by a $n$-independent
constant when $r_{b}=n$.

Additionally, the Noether’s prepotential for Einstein-AdS spaces becomes

$\displaystyle
q^{\mu\nu}=-\kappa\ell^{2}\left(W^{\mu\nu}{}_{\lambda\rho}\pm\star
W^{\mu\nu}{}_{\lambda\rho}\right)\nabla^{\lambda}\xi^{\rho}.$ (46)

Therefore, the Noether charge vanishes identically for (anti-)self dual
solutions as well. This, in turn, implies that the total mass and entropy are
zero for Taub-NUT-AdS. In other words, the role of the Chern-Pontryagin term
with (43) is to set Taub-NUT-AdS solution as the background reference.

The entropy for bolt obtained through the Wald’s formula in Eq. (11) via the
Noether’s prepotential (46) yields

$\displaystyle S_{\rm Bolt}$ $\displaystyle=\frac{16\pi^{2}\kappa
n}{r_{b}^{2}}\Big{[}2\left(r_{b}-n\right)\left(4r_{b}^{2}+5r_{b}n+3n^{2}\right)$
$\displaystyle+\frac{1}{\ell^{2}}\left[3\left(r_{b}+3n\right)\left(r_{b}-n\right)^{2}\left(2r_{b}^{2}+nr_{b}+n^{2}\right)+\ell^{4}\left(2r_{b}+n\right)\right]\Big{]},$
(47)

which, in contrast to (34)-(38), when $r_{b}=n$ is independent on $n$.

It is straightforward to check that the entropy (47) can be obtained from the
thermodynamic relation (36) using the renormalized action (45). The
thermodynamic mass for bolt, on the other hand, is obtained from standard
methods, yielding

$\displaystyle\hat{M}_{\rm Bolt}$ $\displaystyle=\frac{\partial I_{\rm
Bolt}}{\partial\beta_{\tau}}=\frac{8\pi\kappa\left(r_{b}-n\right)^{2}}{r_{b}}\left[1+\frac{\left(r_{b}-n\right)\left(r_{b}+3n\right)}{\ell^{2}}\right]=M_{\rm
Bolt}-M_{\rm NUT},$ (48)

where $M_{\rm Bolt}$ and $M_{\rm NUT}$ are the charges given by (37). The mass
(48) is equivalent to the definition of Ref. Araneda  _et al._ (2016) in terms
of the Noether charge when $\vartheta=+1$.

###  III.3 Boundary stress tensors

The topological renormalization method is consistent with holographic
renormalization Miskovic and Olea (2009), leading to reproduce the boundary
stress tensor. The inclusion of the Chern-Pontryagin term in the gravity
action on AlAdS spaces modifies the result one obtains by means of holographic
renormalization for the dual theory stress tensor $T_{ab}$. In fact, for
$\vartheta=\pm 1$ one obtains999The coefficient of the second term in (49)
acquires an imaginary $i$ factor in the Lorentzian case, due to the presence
of the Levi-Civita pseudotensor in the definition of $C_{ab}$, Eq. (50). Our
conventions on the Wick rotation from Euclidean to Lorentzian are $n\to in$,
$\tau\to i\tau$ and $C_{ab}\to-iC_{ab}$.

$T^{\pm}_{ab}=T_{ab}\mp\frac{\ell^{2}}{8\pi G}C_{ab}\,,$ (49)

where $a,b=0,1,2$, $G$ is the four-dimensional Newton constant and we
introduced the Cotton tensor

$C_{ab}=\sqrt{-g^{(0)}}\varepsilon_{a}{}^{cd}\nabla_{c}\left(R^{(0)}_{db}-\frac{1}{4}R^{(0)}g^{(0)}_{db}\right).$
(50)

Here $g^{(0)}_{ab}$ is the boundary metric while $R^{(0)}_{ab}$ and $R^{(0)}$
its Ricci tensor and scalar curvature, respectively. The tensor $T^{\pm}_{ab}$
is the full stress tensor, which consists of the sum of the holographic part
of standard Balasubramanian-Kraus tensor $T_{ab}$ and the Cotton tensor
$C_{ab}$, with a specific relative coefficient.101010The coupling of tensor
(49) would result in a boundary model of Topologically Massive Gravity Deser
_et al._ (1982). This is related to fluid interpretation of the dual stress
tensor (named “reference tensor”) as well as to the so-called stress tensor /
Cotton tensor duality, cf. Leigh and Petkou (2007); Mansi  _et al._ (2009a,
b); Bakas (2009); Leigh  _et al._ (2012a, b); Miskovic and Olea (2009). We
will not discuss this in extenso here as it lies beyond the scope of the
present paper. Here, we focused on the AlAdS sector; more precisely, on the
thermodynamics of the Misner strings of both Taub-NUT-AdS and Taub-Bolt-AdS
solutions as it is described by the topological renormalization method.

The Cotton tensor in (49) is the parity odd part of $T^{\pm}_{ab}$, while the
Balasubramanian-Kraus piece is parity even. This implies that a parity
transformation results in an interchange $T^{+}_{ab}\leftrightarrow
T^{-}_{ab}$. We discuss the parity transformation below, after writing the
explicit expressions for the tensors; see (52)-(53). Let us just mention here
that the presence of the NUT charge $n$ in the boundary metric, being this
charge parity odd, makes natural to consider in the boundary theory both
components in (49).

Let us now evaluate the full stress-tensor $T^{\pm}_{ab}$ on the Taub-
NUT/Bolt-AdS metrics. To do that, it is convenient to rewrite the metric (20)
by performing the Misner change of coordinate $\tau\to\tau-2n\phi$ followed by
the inversion $\phi\to-\phi$. This yields

$\displaystyle\text{d}s^{2}$
$\displaystyle=f(r)(\text{d}\tau+4n\sin^{2}({\theta}/{2})\,\text{d}\phi)^{2}+\frac{\text{d}r^{2}}{f(r)}+(r^{2}-n^{2})\left(\text{d}\theta^{2}+\sin^{2}\theta\,\text{d}\phi^{2}\right).$
(51)

Evaluating the Balasubramanian-Kraus boundary tensor on this metric
yields111111The minus in the $T_{\tau\tau}$ component makes the fluid energy
density positive in Lorentzian signature.

$T_{ab}=-\frac{M}{8\pi\ell^{2}}\,\left(\begin{array}[]{ccc}{2}&0&{-8n\sin^{2}\frac{\theta}{2}}\\\
0&-\ell^{2}&0\\\
-8n\sin^{2}\frac{\theta}{2}&0&{32n^{2}\sin^{4}\frac{\theta}{2}}\,-\ell^{2}\,\sin^{2}\theta\\\
\end{array}\right),$ (52)

where $a,b={\tau,\theta,\phi}$. The boundary normalized Hamiltonian Killing
vector, interpreted in fluid/gravity as the fluid velocity, is
$u=\partial_{\tau}$; in Euclidean signature this vector is spacelike. This
ensures that in the flat limit this vector becomes null. On the other hand,
the Cotton tensor evaluated on (51) reads

$C_{ab}=-{\frac{n}{\ell^{4}}\left(1-\frac{4n^{2}}{\ell^{2}}\right)}\,\left(\begin{array}[]{ccc}{2}&0&{-8n\sin^{2}\frac{\theta}{2}}\\\
0&-\ell^{2}&0\\\
{-8n\sin^{2}\frac{\theta}{2}}&0&{32n^{2}\sin^{4}\frac{\theta}{2}}\,-\ell^{2}\,\sin^{2}\theta\\\
\end{array}\right).$ (53)

The Cotton energy density is then computed to be
$c=\ell^{4}C_{ab}u^{a}u^{b}=-2n(1-4{n^{2}/\ell^{2}})$. We see here also that
something special happens at the special points $n=\pm\ell/2$, for which the
Cotton tensor vanishes as for $n=0$.

Expressions (52) and (53) satisfy the simple relation

$T_{ab}=\frac{M\ell^{4}}{8\pi n(\ell^{2}-4n^{2})}\,C_{ab}\,,$ (54)

which is manifestly compatible with the stress-tensor / Cotton tensor duality
referred to above. The particular solution (20) that is self-dual corresponds
to the Taub-NUT-AdS spacetime, $MG=n(1-4{n^{2}/\ell^{2}})$, and, in the
Euclidean theory, the complete boundary tensor evaluated on this solution
yields

$T^{+}_{ab}\,\big{|}_{\text{NUT}}=0,$ (55)

and so vanishing energy (respectively, $T^{-}_{ab}\,|_{\text{NUT}}=0$ for the
anti-self-dual solution with NUT charge $-n$). The vanishing of (55) is
consistent with the fact that the mass (48) is measured with respect to NUT
spacetime as a reference background. The charge computation in terms of the
Brown-York stress tensor would yields zero results if
$MG=n(1-4{n^{2}/\ell^{2}})$ is satisfied.

Going back to the question of parity (a)symmetry, we observe that expressions
(52) and (53) realize the fact that, under parity transformation, we get the
map $T^{\pm}_{ab}\to T^{\mp}_{ab}$. In particular, $T^{-}_{ab}=0$ for
$MG=n(4{n^{2}/\ell^{2}}-1)$. The behavior of $T_{ab}$ and $C_{ab}$ under
parity can be immediately observed from the fact that, while the component
$T_{\tau\phi}$ is linear in $n$, the component $C_{\tau\phi}$ is proportional
to a polynomial of even powers of $n$. This is related to the fact that the
change $(\tau,\phi,n)\to(+\tau,-\phi,-n)$ realizes the parity transformation;
it leaves invariant $T_{ab}$ while changes the sign of $C_{ab}$. The same
argument holds for the transformations $(\tau,\phi,n)\to(-\tau,-\phi,+n)$ and
$(\tau,\phi,n)\to(-\tau,+\phi,-n)$. In the fluid interpretation of the
boundary theory Caldarelli  _et al._ (2011); Kalamakis  _et al._ (2020), the
NUT charge is a monopolar source of vorticity. This is reminiscent of a
magnetic monopole. Thence, the stress tensor / Cotton tensor duality is the
fluid analog of electric / magnetic duality. Therefore, once again, the
presence of $n$ in the boundary metric makes natural to consider both pieces
in (49).

##  IV Conclusions

Let us summarize our results: in this paper, we have considered the method of
topological renormalization in AlAdS spaces. The latter consists in adding to
the gravitational action a topological term that, while suffices to render the
Euclidean action finite and the variational principle well posed, provides a
natural definition of the renormalized Noether charges. We used this
prescription to work out the thermodynamics of AlAdS spacetimes, focusing on
the physical properties of the Misner strings of both the Taub-NUT-AdS and
Taub-Bolt-AdS solutions. This enabled us to correctly compute the contribution
of the Misner string to the gravitational entropy by treating on the same
footing the AdS and AlAdS sectors. We also investigated the effects of
introducing the Chern-Pontryagin topological invariant in the gravity action,
namely a parity odd dimension-four operator whose role is setting the Taub-
NUT-AdS geometry as the reference background. We discussed how the presence of
topological terms in the gravitational action contributes to the holographic
stress tensor upon a suitable asymptotic expansion of the fields. While the
relation between topological and holographic renormalization had previously
been studied, our discussion here successes in treating the AdS and AlAdS
sectors in equal footing as well as including the parity odd contributions.
These contributions are responsible for the appearance of the Cotton piece in
the boundary total stress tensor. This augmented stress tensor then
identically vanishes on (anti-)self-dual backgrounds.

The method employed here is generally covariant and could be adapted to other
black hole solutions. In particular, it would be worth pursuing it for other
AlAdS solutions, like accelerating black holes (AdS $C$-metric) or even more
general Robinson-Trautmann spacetimes, whose thermodynamics recently received
renewed interest Appels  _et al._ (2016); Anabalón  _et al._ (2019). We plan
to go back to this problem in future work with the intention to studying
further the role of defects: the presence of the Misner string can be seen as
a boundary defect. Studying this could allow us to make contact with the
intertwining between renormalized volume and area in asymptotically hyperbolic
spaces Anastasiou  _et al._ (2018). Furthermore, the defect corresponds to
introducing corners in the boundary, so it would be interesting to connect our
work with recent discussions on corners Freidel  _et al._ (2020a, b, c). Here,
we focused on the entropy; however, the Noether-Wald’s formalism is well-
suited to study general asymptotic charges. Asymptotic charges have recently
been studied for Taub-NUT/Bolt in locally flat spacetime in relation to the
magnetic counterpart of BMS charges, see e.g. Godazgar  _et al._ (2019); Kol
and Porrati (2020). This raises the interesting question of the Ricci flat
limit of our results, in the spirit of Ciambelli  _et al._ (2018); Compère
_et al._ (2019). Finally, while the total boundary stress tensor
$T^{\pm}_{ab}$ is complex in Lorentzian signature, in Euclidean signature it
is real. Thus, unveiling its microscopic properties in the boundary CFT is
challenging but in principle possible. In particular, we look forward to a
more comprehensive understanding of the microscopic structure of the boundary
dual of Taub-NUT/Bolt-AdS spaces, and its behavior under parity.

###### Acknowledgements.

We thank D. Flores-Alfonso, R. G. Leigh, J. Oliva and A. C. Petkou for
valuable exchanges and discussions. L. C. is supported by the ERC Advanced
Grant High-Spin-Grav. The work of C. C. is supported by FONDECYT No 11200025\.
J. F. acknowledges the financial support of ANID through the Fellowship
22191705. The work of G. G. is supported by CONICET grant PIP 1109-2017. The
work of R. O. is supported by FONDECYT No 1170765\.

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# Reframing the Brain Age Prediction Problem to a More Interpretable and
Quantitative Approach

Neha Gianchandani  Mahsa Dibaji  Mariana Bento  Ethan MacDonald  Roberto Souza

###### Abstract

Deep learning models have achieved state-of-the-art results in estimating
brain age, which is an important brain health biomarker, from magnetic
resonance (MR) images. However, most of these models only provide a global age
prediction, and rely on techniques, such as saliency maps to interpret their
results. These saliency maps highlight regions in the input image that were
significant for the model’s predictions, but they are hard to be interpreted,
and saliency map values are not directly comparable across different samples.
In this work, we reframe the age prediction problem from MR images to an
image-to-image regression problem where we estimate the brain age for each
brain voxel in MR images. We compare voxel-wise age prediction models against
global age prediction models and their corresponding saliency maps. The
results indicate that voxel-wise age prediction models are more interpretable,
since they provide spatial information about the brain aging process, and they
benefit from being quantitative.

Brain Age Prediction, Interpretability, Voxel-level brain age

##  1 Introduction

Researchers have hypothesized that the brain age of healthy subjects should
match their corresponding chronological ages. Increased brain age compared to
chronological age is an important indicator of brain health, making brain age
prediction a widely explored research area. Most brain age prediction work
focuses on predicting a ‘global’ brain age index that reflects the maturity
level and age of the brain. The global brain age index has been shown to be an
effective biomarker to assess the aging process as well as to understand
structural changes in the brain in the presence of neurological disorders
(Cole & Franke, 2017; Wang et al., 2019). Global brain age has most widely
been estimated from T1-weighted Magnetic Resonance (MR) volumes using Deep
Learning (DL) techniques (Cole & Franke, 2017; Tanveer et al., 2023; Jónsson
et al., 2019), as a regression task.

Figure 1: (a) Traditional paradigm where a brain age prediction model is used
to predict a global brain age value for the whole brain. (b) New proposed
paradigm where a voxel-level brain age prediction model assigns different
brain ages to each region of the brain. At the most granular level, each voxel
corresponding to brain tissue in the image can be assigned a brain age.

Predicting the brain age is important to study normal brain aging and
prospective cognitive decline, but understanding and interpreting these
findings, and uncovering the black-box nature of these methods is an even more
relevant task. These models aim to detect brain regions or biomarkers that
should be analyzed in greater detail, leading to personalized diagnosis and
decision-making. Efforts have been made to interpret the DL models with
techniques based on saliency maps, such as Gradient-weighted Class Activation
Mapping (Grad-CAM) (Bermudez et al., 2019; Yin et al., 2023; Selvaraju et al.,
2017), and occlusion-based techniques (Bintsi et al., 2021; Zeiler & Fergus,
2014). Saliency map techniques focus on creating visualizations to depict the
contribution levels of each pixel in the decision-making process, whereas
occlusion-based techniques aim to identify regions that are most important to
make a particular decision by hiding regions in the input and observing their
impact on model performance. The worse the model performs when hiding a
certain feature, the higher its importance. There are other interpretability
techniques like Layer Wise Relevance Propagation (Bach et al., 2015) and
SHapley Additive exPlanations (SHAP) (Lundberg & Lee, 2017), however, to the
best of our knowledge, they have not been utilized to explain brain age
prediction models.

At a high level, the aforementioned techniques provide an understanding of how
DL models learn and explain the relationship between the features and the
models’ decision making, hence, providing interpretability to DL models, i.e.,
unboxing the black-box. Existing interpretability techniques help with
understanding the global brain age prediction models by accessing the DL
model’s decision-making. Such techniques explain what features in the model
input contribute to the decision-making process (i.e. to predict global brain
age) providing a spatial-level analysis of the problem, relevant to identify
specific brain regions of interest, biomarkers, and abnormalities related to
aging.

It is important to highlight that most saliency-based interpretability
techniques were initially proposed to explain classification models, producing
heat maps to a specific class. The translation of such techniques for
regression tasks is not straightforward.

Figure 2: Proposed voxel-level brain age prediction model architecture. The
model has a U-Net backbone and follows a multi-output design with three
outputs: (i) Segmentation of Gray Matter, White Matter and CSF, (ii) Voxel-
level brain age prediction, and (iii) Global-level brain age prediction.

Figure 3: Global age prediction model architecture. The backbone is adopted
from (Peng et al., 2021), and the output head is modified to treat global age
prediction as a regression problem rather than a classification problem as
done in the original research.

In this article, we evaluate our recently proposed voxel-level brain age
prediction model (Gianchandani et al., 2023) from an interpretability
perspective. Interpretability has been defined differently in varying contexts
(Doshi-Velez & Kim, 2017; Kim et al., 2016; Miller, 2019), however, all
definitions aim to achieve a common goal: to satisfy human curiosity (Miller,
2019) and in the machine learning context, to make the modeling pipeline (from
feature extraction to decision making) less of a black box, and more
transparent. An alternative to utilizing interpretability methods to explain
the feature extraction by previously trained age prediction models, we propose
to redefine how we approach the research problem. The aim of predicting brain
age is to understand how the brain ages in healthy compared to diseased
brains. We further want to understand how various regions contribute to the
decision-making process reflecting more on the spatial aging processes in the
brain. Instead of approaching it as a global age prediction problem, we
propose to approach it as an image-to-image regression problem where we
predict brain age at a voxel level. So far to the best of our knowledge, only
(Popescu et al., 2021) have attempted a voxel-level brain age prediction
approach. The authors use a U-Net-like (Ronneberger et al., 2015) architecture
to predict voxel-level brain age and achieve a Mean Absolute Error (MAE) of
$9.94\pm 1.73$ years. However, the approach is not discussed from an
interpretability point of view.

A voxel-level approach will enable a spatial-level analysis of the brain aging
process by assigning a brain age prediction to each voxel of the brain.
Individual predictions for each volume unit not only allow us to analyze the
aging process at a more fine-grained level but also provides a quantitative
visualization that reflects how different regions of the brain are aging. A
brain voxel in the image that is assigned an increased or decreased brain age
index can be explained by the contribution of the voxel in the decision-making
process by the DL model. In this article, we propose a voxel-level age
prediction model as a new approach to interpretability. An overview of global
age versus voxel-level age prediction outputs is described in Fig. 1.

We further compare our new proposed method to the existing methods for
interpretability. Grad-CAM (Selvaraju et al., 2017), Occlusion Sensitivity
maps (Zeiler & Fergus, 2014) and SmoothGrad (Smilkov et al., 2017) are
utilized to explain a publicly available state-of-the-art global brain age
prediction model, and the interpretations are discussed and compared with our
proposed voxel-level brain age prediction model.

We acknowledge the existing state-of-the-art methods for brain age prediction
and emphasize that the focus of this article is not to propose state-of-the-
art brain age prediction models, but to contrast existing interpretability
techniques with our proposed approach to the brain age prediction problem to
uncover the black box and to better understand the predictions.

##  2 Materials and Methods

###  2.1 Data

We used 3D T1-weighted MR scans from the publicly available Calgary-Campinas
dataset (Souza et al., 2018). We will refer to this dataset as $D_{cc}$
hereafter in this article. All data corresponds to presumed healthy subjects.
$D_{cc}$ has 359 samples aged 29-80 years (mean age of 53.47$\pm$7.84) with a
male:female sex ratio of 49:51 percent. The data was acquired on three
different scanners (Philips, Siemens, and General Electric), each at two
magnetic field strengths (1.5 T and 3 T). Skull-stripping masks, as well as
tissue segmentation masks (Gray Matter (GM), White Matter (WM), and
Cerebrospinal Fluid (CSF)), are also publicly available with the dataset.

###  2.2 Data Preprocessing

For the voxel-level brain age prediction model, the scans are reoriented to a
standard template (MNI152 (Fonov et al., 2009)) using the FSL (Jenkinson et
al., 2012) utility ‘reorient2std’ to ensure consistency across the dataset. We
adjusted each sample’s intensity to fall within the range of 0 to 1. Any other
kind of preprocessing steps, such as registration is avoided for this model as
the predictions are to be done at a voxel level, and we want to ensure that
the input features remain untouched and the brain structures hold the shape
and intensity values as in the original reconstructed T1-weighted image.

For the global age prediction model, the MR scans are linearly registered with
6 degrees of freedom using FMRIB Software Library’s (FSL) FLIRT (FMRIB’s
Linear Image Registration Tool) command (Jenkinson & Smith, 2001), which
allows for rotation and translation, keeping the shape of the brain consistent
to avoid loss/change in input data features. The registration step was seen to
improve the performance of the global age prediction model and hence, was
included in the pipeline.

###  2.3 Proposed Model Architectures

####  2.3.1 Voxel-level brain age prediction model

The proposed model follows the 3D U-Net (Ronneberger et al., 2015)
architecture backbone (Fig. 2). The model has an encoder network (left) and
decoder network (right) joined together with a bottleneck (center bottom),
making an U shape. The input is downsampled at each level of the encoder as
features are learned, whereas the feature map is interpolated back to the
original input size iteratively at each level of the decoder. Skip connections
are used to connect the encoder and the decoder at each level to allow for
feature re-usability as well as help with the vanishing gradient problem.
Batch normalization is used post convolution layers in the encoder network.

The model follows a multitask architecture with three outputs for which the
features are learnt simultaneously. This forces the model to learn similar
features for the three tasks. The three tasks defined for the model are: (i) A
segmentation task to segment GM, WM, CSF. (ii) Voxel-level brain age
prediction task. The U-Net architecture helps in maintaining output size to be
the same as the input so as to obtain voxel-level brain age prediction for
each voxel in the input age. A Rectified Linear Unit (ReLU) activation is used
to ensure positive age predictions. (iii) Global-level brain age prediction
task, which is computed from the bottleneck features of the model and is a
scalar age prediction for each volume. This task closely resembles the output
from the global age prediction model described in Section 2.3.2.

Task (i) and (iii) are helper tasks that aid the model in learning accurate
features for the voxel-level age prediction task. Adding the segmentation task
is especially useful as it pushes the model to learn structural features for
the voxel-level age prediction task ignoring other unnecessary information
present in the input images. The global-level brain age can be thought of as a
prerequisite and simpler version of the voxel-level brain age prediction task.

####  2.3.2 Global Brain Age prediction model

We adapt the architecture for the global brain age prediction model from a
state-of-the-art Simple Fully Convolutional Neural network (SFCN) proposed in
(Peng et al., 2021; Gong et al., 2021). The authors treat the brain age
prediction task as a soft classification task where the model predicts a
Gaussian Probability distribution centered at the ground truth (chronological
age) instead of a scalar brain age index. The model is lightweight as it is
made up of 7 convolution blocks where the input is down-sampled after each
convolution layer in the first five blocks with $3\times 3\times 3$
convolutions, followed by a $1\times 1\times 1$ convolution block and a
classification head. Batch normalization is also used to ensure a smooth
training process. We modify the architecture and retain the convolution
blocks, but treat it as a classical regression problem (Fig. 3).

###  2.4 Loss Function

####  2.4.1 Voxel-level brain age prediction model

A combination loss function is used to train the voxel-level brain age
prediction model. The loss function is a weighted sum of three loss terms, one
corresponding to each task. The Soft Dice Score (Milletari et al., 2016) is
used for the segmentation task, and MAE at the global and voxel level is used
for the brain age prediction task.

The weighting terms ($\alpha$, $\beta$, and $\gamma$) used are initialized and
changed as the model trains in a way that all tasks are given equal importance
throughout the training process. The loss function is described in equation 1.

$\begin{split}L_{overall}=\alpha Dice_{loss}+\beta MAE_{global}+\\\ \gamma
MAE_{voxel}\end{split}$ (1)

####  2.4.2 Global Brain Age prediction model

We use the MAE as the loss function for the global age prediction model. The
decision to use MAE as the loss function was reinforced after running
experiments with Mean Squared Error (MSE) as the loss function. Experiments
showed that MAE performed significantly better leading to an improved training
trajectory with a lower validation loss and a smoother training/validation
curve.

###  2.5 Training Methodology

####  2.5.1 Voxel-level brain age prediction model

We train the model for 300 epochs with an initial learning rate of 0.001,
which reduces every 70 epochs by a multiplicative factor of 0.5. A batch size
of 8 is used for training where each input sample is a randomly cropped patch
of size $96\times 96\times 96$ from the T1-weighted volume. A single crop is
randomly obtained from each volume ensuring significant brain volume in each
crop while exposing the model to the same brain region from various
perspectives. 50% of the cropped patches are rotated by 15∘ as an augmentation
step on-the-fly. To prevent the model from solely learning to predict global
brain age for each voxel, we introduce small noise to the ground-truth labels
before calculating the loss. For voxel-level age prediction task, instead of
the ground truths having the the same global age at each voxel, we introduce
small noise (in the range of [-2,2]) to ensure the models learns variations in
age across different voxels (or regions) without significantly impacting the
error.

####  2.5.2 Global Brain Age prediction model

In the training process for the global brain age prediction model, we ran the
model for 50 epochs, starting with a learning rate of 0.0001. As the training
progressed, learning rate was decreased by half every 20 epochs. We trained
with a batch size of 8, and adjusted each sample’s intensity to fall within a
scale of 0 to 1. We also implemented an on-the-fly augmentation step, rotating
50% of the samples by 15∘. This approach aimed to increase the model’s
robustness by providing a diverse set of sample variations throughout the
training process.

Figure 4: Traditional interpretability methods (left to right- Grad-CAM,
Occlusion Sensitivity maps, and SmoothGrad) implemented on global age
prediction model and voxel-level brain PAD masks obtained from voxel-level
brain age prediction model.

###  2.6 Interpretabily Methods

There are several techniques employed to explain the decision-making process
of a deep-learning model, such as Grad-CAM (Selvaraju et al., 2017). This
method uses the gradients flowing into the final convolutional layer to
produce a coarse localization map, highlighting the image regions important to
prediction. Its model-agnostic property makes it applicable across a wide
range of Convolutional Neural Network (CNN)-based models, without requiring
retraining. However, the main limitation of Grad-CAM is its coarse
localization due to the low spatial resolution of deeper convolutional layers,
which can sometimes limit its interpretability. To counteract this limitation,
we employed a method that averages two maps—one from an earlier convolutional
layer as the target layer and one from the final layer—to blend detailed
features with high-level representations for a more comprehensive
understanding (Morbidelli et al., 2020). It is important to acknowledge that
the inputs to the two layers considered for generating the heatmaps are
different in terms of the input feature maps as well as dimensions.
Aggregating two feature maps, originating from layers at distinct depths in
the model helps to smooth out inconsistencies or noise that may exist in the
heatmap from the final convolutional layer. It also helps in obtaining
informative heatmaps by encompassing both relatively high-level and low-level
features present at different depths in the network (McAllister et al., 2020).

Another interpretability technique, Occlusion Sensitivity (Zeiler & Fergus,
2014), provides an interpretability approach that systematically occludes
portions of the input image to observe changes in model output, thereby
producing a heat map of the input’s most influential regions. The strength of
this technique lies in its simplicity and general applicability; it can be
applied to any model predicting based on an image input, even black box
models, without requiring access to model gradients. Nevertheless, the
technique is not without drawbacks. Specifically, Occlusion Sensitivity is
computationally intensive, necessitating a rerun of the model for every
occluded version of the image, which makes it somewhat inefficient for larger
images or complex models. Additionally, the results can be influenced by the
size and shape of the occluding patch, making it important to choose these
parameters carefully.

Finally, SmoothGrad (Smilkov et al., 2017) offers another approach to
interpretability. In contrast to the previous techniques, SmoothGrad aims to
reduce noise in gradient-based saliency maps by averaging the gradients of
multiple noisy versions of the same input image. Consequently, it often yields
less noisy saliency maps than single input-based methods, highlighting noise-
robust patterns. However, SmoothGrad requires multiple forward passes through
the model as well as repeated gradient calculation to obtain smoother saliency
maps for each input, which increases computational demands. Similarly to
Occlusion Sensitivity, the choice of appropriate noise level may prove
challenging and require further experimentation or tuning.

For our experiments, we implement interpretability methods using MONAI
(Cardoso et al., 2022), which is a PyTorch-based framework that provides
built-in functions for implementing various interpretability techniques.

##  3 Results

The voxel-level brain age prediction model achieved an MAE of $5.96\pm 3.75$
years on the $D_{cc}$ test set (n=30 samples). The global brain age prediction
model achieved an MAE of $6.56\pm 4.04$ years on the same test samples (see
Table 1).

Table 1: Results for voxel-level and global brain age prediction models on the
$D_{cc}$ test set
Model | Test Set |  MAE$\pm$S.D |
---|---|---|---
Voxel-level model | $D_{cc}$ |  5.96$\pm$ 3.75 |
Global model | $D_{cc}$ |  6.56$\pm$ 4.04 |

For the voxel-level age prediction model, we present results using MAE
averaged across all voxels of the brain for simplicity. It is not feasible to
report voxel-level MAE for all (millions) of the brain voxels individually.
However, for visualization, we use predicted age difference (PAD) masks that
show the difference between the predicted brain age and the chronological age
at the level of each voxel. Blue color indicates brain regions that look
younger than chronological age and red points to older-looking brain regions.
PAD masks can be an excellent way to visualize brain maturity levels from
voxel-level brain age predictions. The masks also provide an additional level
of quantification to explain the outputs of the DL model.

For the global age prediction model, we report the MAE over the test set and
also present the results of three different interpretability techniques (Fig.
4).

##  4 Discussion

Figure 4 shows the comparison of traditional interpretability methods to
voxel-level PAD masks. Each row represents different interpretability outputs
for the same test subject. Grad-CAM heatmaps show the important regions that
contributed to the decision-making process (to predict global brain age) using
a red-yellow-blue gradient colormap with red regions being the biggest
contributors and blue regions being the smallest contributors or least
important. However, as can be observed in Fig 4, Grad-CAM heatmaps are coarse
and prone to up-sampling errors. Grad-CAM maps are also not comparable across
different samples as they are hard to quantify, this is due to the origin of
the heatmap intensity values which come from the relative strength of
gradients for a particular input image.

Similarly, the occlusion sensitivity maps for classification problems show the
most and least important regions for classifying a specific class, however, in
the context of a regression problem such as brain age prediction, red points
refer to regions that make the model overestimate the brain age (prediction
$>$ ground truth) and blue points refer to regions that make the model
underestimate brain age (prediction $<$ ground truth). This means that all
regions in red and blue contribute to the brain age prediction decision in
some way, and the white regions are the least contributing.

Saliency maps created using SmoothGrad highlight the areas that significantly
influence the model’s output. These crucial regions are typically marked in
red, while the areas having a minimal effect are illustrated in blue. As
observed, these maps offer more detailed insights compared to Grad-CAM,
seemingly highlighting important regions with more precision. This enhanced
level of detail could be attributed to the fact that SmoothGrad averages over
multiple versions of the same input with added noise. Similar to Grad-CAM,
SmoothGrad is also a gradient-based technique, it relies on the quantitative
nature of the gradients, and how changing the input affects the gradients in
the model. Thus, it reflects the relative importance of different regions in
the input image (Brain MR).

Overall, while distinct differences exist in the maps generated by the three
techniques, a general agreement can be seen regarding the key regions utilized
in predicting global age. While the traditional interpretability methods do a
good job of providing a high-level understanding of important regions, they
are based on relative scores for a specific input sample, and not absolute
quantitative measurements which can be used to directly compare different
samples.

Figure 5: Segmentation results using the voxel-level brain age prediction
model on one sample. The model achieved a mean dice score of 84.25% on the
test set.

The voxel-level PAD masks on the other hand are used to visualize individual
predictions for each voxel (region) of the brain. The PAD values embedded in
the masks quantify the difference between the brain age prediction and
chronological age (in years), consequently making the PAD masks comparable
across different samples. Owing to the multitask design of the model
architecture, the model is forced to learn features that can be repurposed for
all three tasks. The addition of the brain tissue segmentation task ensures
that the model learns structural features within the brain region that will be
used for the segmentation as well as the voxel-level brain age prediction
task. We achieve a Dice Score of 84.25% on the $D_{cc}$ test set indicating a
high overlap between predicted segmentation and ground truth for GM, WM, and
CSF, hence accurate segmentations (Fig. 5). This is only possible if correct
structural features are learned during the feature extraction process while
training the model. This further confirms that structural features within the
brain region are contributing to the voxel-level age prediction task. It can
also be hypothesized that big PAD values can be partially explained by the
presence of underlying structural anomalies that have not evolved into a
disease suggesting the contribution of the region in predicted voxel-level
brain age. Fig. 4 shows PAD masks of healthy subjects from our test set,
research has shown that the brain aging process varies across different
regions of the brain. Studies have also shown that each healthy brain is
unique and follows a different spatial aging trajectory (Raz et al., 2005;
Scahill et al., 2003) and this can be observed in the PAD masks. The
underlying structural features of the brain cause the predicted age difference
of a voxel to be non-zero PAD and this varies spatially in the brain.

Voxel-level PAD maps also provide an increased resolution at the level of each
voxel, unlike traditional interpretability methods which are usually noisy at
such granular level (Liu et al., 2021). Saliency map-based techniques like
Grad-CAM and even SmoothGrad only indicate the activity of broad regions in
the final output, however, due to the heatmaps undergoing interpolation (for
upsampling), it becomes nonviable to correlate the importance of regions to
specific morphological changes that have caused a particular prediction. Age
prediction at a voxel level can lead to a fine-grained analysis of underlying
changes in the brain. For the purpose of this article, our aim is to compare
how various traditional interpretability methods compare to voxel-level brain
PAD masks, hence, we will base all our comparisons on the voxel-level PAD
maps. However, in the future, to correlate variation observed in PAD maps to
existing research on aging, which often focuses on regions rather than
individual voxels, it can be valuable to average the voxel-level brain age
predictions within the known anatomical regions of the brain to visualize and
study the variation in aging trajectories across regions of the brain. In the
future, it will also be valuable to get clinical feedback to better understand
the aging patterns observed in the voxel-PAD maps. This can help correlate the
spatial variations observed in the brain to existing research on aging at a
regional level.

Occlusion-based sensitivity maps come close to providing similar insights as
voxel-level PAD maps, however, with occlusion sensitivity maps, we only get an
estimation of single regions contributing to over or under-estimation of
overall global brain age. The impact of an occluded region on the output is
hard to quantify as it’s rarely ever one region in the image that contributes
entirely to the final output. The output of DL models is a combination of
multiple features, and occlusion sensitivity maps do not account for the
impact of occluding multiple regions together.

Most traditional interpretability techniques have the advantage of being
utilized for multiple use cases and with different model architectures,
however, they are used as a post-modeling step to assess if the trained model
has learned accurate features. Our proposed model on the other hand is
specific to the brain age prediction problem (as of now, although, can be
extended to other imaging research problems), however, it is a modeling
technique, rather than an additional algorithm to check for interpretability.
It ensures that accurate features are learned to predict brain age and
additionally, also provides spatial information on the brain aging process.

##  5 Conclusion

In this article, we used our recently proposed voxel-level brain age
prediction (Gianchandani et al., 2023), as a step towards interpretability in
the brain age prediction realm. This perspective on brain age prediction is
relatively new and has not been explored from an interpretability point of
view in the past. We also compare the outputs of the proposed voxel-level age
prediction model to existing traditional interpretability methods and reflect
on the differences between them. Through our findings, we show that our
proposed model provides an additional level of interpretability, and fine-
grained analysis of the features used for the decision-making process by the
model and is quantitative in nature.

## Acknowledgements

The authors would like to express our appreciation to the reviewers for their
feedback which helped in refining and strengthening the final version of the
paper. This project is supported by the Hotchkiss Brain Institute, University
of Calgary, Alberta Innovates, the Natural Sciences and Engineering Research
Council of Canada (NSERC), and the Digital Research Alliance of Canada. The
authors would like to thank Denvr Dataworks for not only providing access to
their high-performance supercluster but also for making our compute-intensive
project more sustainable. With their environmentally conscious cloud
infrastructure and user-friendly AI Cloud software, we were able to train our
models efficiently while conserving water and reducing our carbon footprint.
This collaboration allowed us to advance our research in interpretable medical
machine intelligence, creating a positive impact on healthcare while
demonstrating our commitment to sustainable practices.

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