# Euclidean flows, solitons and wormholes in AdS from M-theory

Andrés Anabalón<EMAIL_ADDRESS>Instituto de Física Teórica, UNESP-
Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo
01140-070, SP, Brazil  Álvaro Arboleya<EMAIL_ADDRESS>Departamento de
Física, Universidad de Oviedo, Avda. Federico García Lorca 18, 33007 Oviedo,
Spain  Instituto Universitario de Ciencias y Tecnologías Espaciales de
Asturias (ICTEA), Calle de la Independencia 13, 33004 Oviedo, Spain  Adolfo
Guarino<EMAIL_ADDRESS>Departamento de Física, Universidad de
Oviedo, Avda. Federico García Lorca 18, 33007 Oviedo, Spain  Instituto
Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Calle
de la Independencia 13, 33004 Oviedo, Spain

###### Abstract

Multi-parametric families of non-supersymmetric EAdS4 flows as well as
asymptotically EAdS4 solitons and wormholes are constructed within the four-
dimensional $\textrm{SO}(8)$ gauged supergravity that describes the
compactification of M-theory on $\textrm{S}^{7}$. More concretely, the
solutions are found within the so-called STU-model that describes the
$\textrm{U}(1)^{4}$ invariant sector of the theory. The on-shell action and
gravitational free energy are computed for the regular solutions, the latter
being zero for the wormholes irrespective of supersymmetry. There are special
loci in parameter space yielding solutions with enhanced (super)symmetry.
Examples include a supersymmetric EAdS4 flow with
$\textrm{SO}(4)\times\textrm{SO}(4)$ symmetry dual to a specific real mass
deformation of ABJM on S3 as well as a non-supersymmetric wormhole with
$\textrm{SU}(3)\times\textrm{U}(1)^{2}$ symmetry. The uplift of these examples
to Euclidean solutions of eleven-dimensional supergravity is presented and
their complex nature discussed.

###### pacs:

04.65.+e, 04.50.+h,11.25Mj

##  I Introduction

General Relativity in Euclidean signature has played a central role in
understanding thermodynamical aspects of gravitational systems, the most
prominent example being the black hole thermodynamics Gibbons:1976ue . In the
more modern context of the gauge/gravity correspondence Maldacena:1997re ,
Euclidean solutions of supergravity theories have been utilised to explore
thermodynamical aspects of strongly-coupled field theories in the planar
limit. An example is the holographic evaluation of (the finite part of) the
free energy of a three-dimensional conformal field theory (CFT) placed on S3,
which has been proposed as a measure of the number of its degrees of freedom.
In analogy with the $c$-theorem Zamolodchikov:1986gt , this free energy is
conjectured to obey a monotonicity property under the renormalisation group
(RG) flow, namely, it obeys an $F$-theorem Jafferis:2011zi ; Klebanov:2011gs .

This note is framed within the specific context of the AdS4/CFT3
correspondence. On the gravity side, our playground is the four-dimensional
(Euclideanised) ${\mathcal{N}=2}$ STU-model of Cvetic:1999xp  which describes
the $\textrm{U}(1)^{4}$ invariant sector of the maximal $\textrm{SO}(8)$
gauged supergravity deWit:1982ig . The latter describes the consistent
truncation of eleven-dimensional (11D) supergravity on S7 to its zero mass
sector deWit:1986oxb . On the field theory side, the CFT3 of relevance is the
ABJM theory describing the worldvolume dynamics of a stack of $N$ planar
M2-branes in flat space Aharony:2008ug . Placing the theory on S3 and turning
on $\mathcal{N}=2$ real mass parameters, which modify the assignment of
$\textrm{U}(1)_{\textrm{R}}$ R-charges and break conformality, turns out to
induce RG flows. Within the STU-model, such RG flows were holographically
constructed in Freedman:2013oja  as a three-parameter family of
$\mathcal{N}=2$ Euclidean solutions that preserve the isometries of the S3 in
the bulk and have an S3 boundary. Building upon these results, we will
construct various new classes of Euclidean solutions within the STU-model:
non-backreacted EAdS4 flows with non-trivial scalar fields in the bulk, as
well as backreacted solutions describing singular solitons and regular
wormholes.

Examples of Euclidean four-dimensional solutions with a non-backreacted EAdS4
geometry and non-trivial matter fields have previously been constructed in
Nakayama:2016ydc  and uplifted to M-theory in Nakayama:2016xzs . These
examples involve non-trivial vector fields in the bulk and describe a non-
trivial end point of the RG flow induced by topologically twisted (scalar)
deformations which preserve the scale invariance but not the conformal
invariance of the CFT3 in the UV. Here we will present a three-parameter
family of non-backreacted EAdS4 flows with running scalar fields (instead of
vectors) which are generically non-supersymmetric. For a specific choice of
parameters, the solution becomes supersymmetric and lies within the class
constructed in Freedman:2013oja . We will discuss its holographic realisation
as a specific real mass deformation of ABJM on S3 and present its uplift to
11D supergravity.

In addition to the non-backreacted flows, we also present multi-parametric
families of soliton and wormhole solutions of the (Euclideanised) STU-model
with running scalars in the bulk. While the solitons are always singular,
fully regular wormholes are shown to exist upon judicious choice of the free
parameters. The construction of regular Euclidean wormholes in
string/M-theoretic scenarios has proven a challenging task. Some constructions
for flat-space wormholes have been put forward in the context of
(super)gravity coupled to proper-scalars and pseudo-scalars (a.k.a. axions)
Giddings:1987cg ; Arkani-Hamed:2007cpn , as well as for AdS wormholes in
presence of a cosmological constant or a scalar potential Hertog:2017owm ;
Marolf:2021kjc ; Loges:2023ypl ; Astesiano:2023iql . In all these
constructions, the analytic continuation from Lorentzian to Euclidean
signature becomes subtle as far as the axions are concerned. Fortunately, a
full-fledged construction of the Euclidean STU-model was done in
Freedman:2013oja  and shown to require, amongst other modifications, a
doubling of the scalar degrees of freedom: the three complex scalars $z_{i}$
of the STU-model and their would-be conjugates $\tilde{z}_{i}$ should be
treated as independent fields. This additional freedom in the Euclidean theory
is precisely what allows us to construct regular wormholes for which
$\tilde{z}_{i}\neq z^{*}_{i}$. When uplifted to eleven-dimensional
supergravity, both the 11D metric and the four-form flux become complex-valued
with non-trivial real and imaginary parts. Although this is in principle
allowed in a theory of supergravity, it remains to be seen what the ultimate
fate of these Euclidean wormholes will be in a theory of quantum gravity.

##  II Euclidean supergravity

The STU-model of Cvetic:1999xp  describes the $\textrm{U}(1)^{4}$-invariant
sector of the $\textrm{SO}(8)$-gauged maximal supergravity deWit:1982ig . Its
Euclidean version, which is the relevant one for this work, has been discussed
in full detail in Freedman:2013oja . In the absence of vector fields, this
supergravity describes an Einstein-scalar model involving a set of complex
scalars $z^{i}$ and $\tilde{z}^{i}$ with $i=1,2,3$. The bulk Euclidean
(bosonic) action is given by

$S_{\textrm{bulk}}=\frac{1}{2\kappa^{2}}\int
d^{4}x\,e\left[-\frac{R}{2}+\sum_{i=1}^{3}\frac{\partial_{\rho}z_{i}\partial^{\rho}\tilde{z}_{i}}{\left(1-z_{i}\tilde{z}_{i}\right)^{2}}+V\right]\
,$ (1)

with $2\kappa^{2}=8\pi G_{4}$ and $G_{4}$ being the four-dimensional Newton
constant. The bulk action (1) includes a non-trivial scalar potential

$V(z_{i},\tilde{z}_{i})=g^{2}\left(3-\sum_{i}\frac{2}{1-z_{i}\tilde{z}_{i}}\right)\
,$ (2)

where $g$ is the four-dimensional gauge coupling that is related to the
(inverse) radius of the seven-sphere. Importantly, as emphasised in
Freedman:2013oja , the scalars $z_{i}$ and $\tilde{z}_{i}$ are complex and
independent in the Euclidean theory, and must separately parameterise the
Poincaré unit-disk, namely, $|z_{i}|,|\tilde{z}_{i}|<1$ $\forall i$. The
spacetime metric should also be allowed to be complex in the Euclidean theory,
although we are only considering real metrics in this work.

The bulk action, as specified by (1) and (2), possesses three global
continuous symmetries $\textrm{SO}(1,1)_{\xi_{i}}$ acting as scalings of the
form $z_{i}\rightarrow e^{\xi_{i}}\,z_{i}$ and $\tilde{z}_{i}\rightarrow
e^{-\xi_{i}}\,\tilde{z}_{i}$ with ${\xi_{i}\in\mathbb{R}}$. A proper
understanding of global symmetries in the bulk would require the inclusion of
the vector fields in the STU-model as they couple to the scalars through
kinetic and topological terms. The $\textrm{SO}(1,1)_{\xi_{i}}$ symmetries
would act linearly on the vector fields changing their boundary values and,
therefore, also changing the dual field theory on the boundary. However, we
are not considering vector fields in this work and set them to zero.

We will construct non-conformal flow solutions in the Euclidean STU-model that
involve non-constant scalars. Moreover, we will demand that the spacetime
geometry has an S3 boundary and that the flows preserve the
$\textrm{SO}(4)\sim\textrm{SU}(2)_{l}\times\textrm{SU}(2)_{r}$ isometry group
of S3 in the bulk. The Euclidean metric (in the Fefferman-Graham gauge) is
then of the form

$ds_{4}^{2}=d\mu^{2}+g^{-2}e^{2A(\mu)}d\Sigma_{\textrm{S}^{3}}^{2}\ ,$ (3)

where $\mu\in\mathbb{R}$ is the radial (holographic) coordinate and
$d\Sigma_{\textrm{S}^{3}}^{2}$ is the line element of a three-sphere
$\textrm{S}^{3}$. As a result, the geometry is specified by the metric
function $A(\mu)$ which we take to be real-valued. Moreover, in order to
preserve the S3 isometries, the scalars can only depend on the radial
coordinate, namely, $z_{i}=z_{i}(\mu)$ and $\tilde{z}_{i}=\tilde{z}_{i}(\mu)$.
With this ansatz, the second-order equations of motion that follow from the
bulk action (1) are given by

$\begin{array}[]{rcll}A^{\prime\prime}+g^{2}e^{-2A}+\displaystyle\sum_{i=1}^{3}\dfrac{z_{i}^{\prime}\,\tilde{z}_{i}^{\prime}}{\left(1-z_{i}\tilde{z}_{i}\right)^{2}}&=&0&,\\\\[11.38109pt]
z_{i}^{\prime\prime}+3A^{\prime}z_{i}^{\prime}+2\dfrac{\tilde{z}_{i}(z_{i}^{\prime})^{2}}{1-z_{i}\tilde{z}_{i}}+2g^{2}z_{i}&=&0&,\\\\[11.38109pt]
\tilde{z}_{i}^{\prime\prime}+3A^{\prime}\tilde{z}_{i}^{\prime}+2\dfrac{z_{i}(\tilde{z}_{i}^{\prime})^{2}}{1-z_{i}\tilde{z}_{i}}+2g^{2}\tilde{z}_{i}&=&0&,\end{array}$
(4)

and are obviously invariant under the $\textrm{SO}(1,1)_{\xi_{i}}$ scaling
symmetries of the bulk action.

The Euclidean STU-model admits an $\mathcal{N}=1$ rewriting in terms of a
Kähler potential

$\mathcal{K}=-\sum_{i=1}^{3}\log\left[1-z_{i}\tilde{z}_{i}\right]\ ,$ (5)

and holomorphic superpotentials

$W(z_{i})=g\,(1+z_{1}z_{2}z_{3})\hskip 8.53581pt,\hskip
8.53581pt\tilde{W}(\tilde{z}_{i})=g\,(1+\tilde{z}_{1}\tilde{z}_{2}\tilde{z}_{3})\
.$ (6)

In this language, the bulk action (1) is constructed from $\mathcal{K}$, $W$
and $\tilde{W}$ using standard $\mathcal{N}=1$ formulae adapted to Euclidean
signature (see Freedman:2013oja ; Bobev:2018wbt ), and a set of first-order
BPS equations can be derived by requiring the vanishing of the supersymmetry
variations of the fermions in the model. The BPS equations can be written as
first-order flow equations for the scalars $(z_{i},\tilde{z}_{i})$ and the
metric function $A$, with the $\mathcal{N}=1$ gravitino mass $m_{3/2}\propto
e^{\frac{\mathcal{K}}{2}}(W\tilde{W})^{\frac{1}{2}}$ playing the role of a
scalar superpotential. Using $\mathcal{K}$, $W$ and $\tilde{W}$ in (5) and
(6), the set of BPS equations reads

$\begin{array}[]{rcll}-1+\dfrac{e^{2A}}{g^{2}}(A^{\prime})^{2}&=&e^{2A}\dfrac{\left(1+z_{1}z_{2}z_{3}\right)\left(1+\tilde{z}_{1}\tilde{z}_{2}\tilde{z}_{3}\right)}{\prod_{i=1}^{3}(1-z_{i}\tilde{z}_{i})}&,\\\\[14.22636pt]
\dfrac{e^{A}}{g}\dfrac{\left(1+\tilde{z}_{1}\tilde{z}_{2}\tilde{z}_{3}\right)z_{i}^{\prime}}{1-z_{i}\tilde{z}_{i}}&=&\left(\pm
1-\dfrac{e^{A}}{g}A^{\prime}\right)\left(z_{i}+\dfrac{\tilde{z}_{1}\tilde{z}_{2}\tilde{z}_{3}}{\tilde{z}_{i}}\right)&,\\\\[11.38109pt]
\dfrac{e^{A}}{g}\dfrac{\left(1+z_{1}z_{2}z_{3}\right)\tilde{z}_{i}^{\prime}}{1-z_{i}\tilde{z}_{i}}&=&\left(\mp
1-\dfrac{e^{A}}{g}A^{\prime}\right)\left(\tilde{z}_{i}+\dfrac{z_{1}z_{2}z_{3}}{z_{i}}\right)&,\end{array}$
(7)

where the upper (lower) choices of sign correspond to using Killing spinors
that are proportional to the left-invariant (right-invariant) Killing spinors
on the $\textrm{S}^{3}$ Freedman:2013oja . Note that both sign choices are
related by the exchange $z_{i}\leftrightarrow\tilde{z}_{i}$. The BPS equations
(7) are generically not invariant under the $\textrm{SO}(1,1)_{\xi_{i}}$
global symmetries of the bulk action unless two out of the three scalars
$z_{i}$ (equivalently for $\tilde{z}_{i}$) are set to zero. While the general
flow solutions we will present in this note are non-supersymmetric, and
therefore satisfy (4) without satisfying (7), we will identify a particular
case for which the BPS equations (7) are additionally satisfied. This
supersymmetric case precisely turns off two out of the three complex scalars,
i.e. $z_{2}=z_{3}=\tilde{z}_{2}=\tilde{z}_{3}=0$, so that the remaining
$\textrm{SO}(1,1)_{\xi_{1}}$ scaling symmetry is also a symmetry of the BPS
equations in the bulk.

All the flow solutions we will construct asymptote to the maximally (super)
symmetric EAdS4 vacuum of the $\textrm{SO}(8)$ supergravity. This vacuum
solution uplifts to the $\textrm{EAdS}_{4}\times\textrm{S}^{7}$ Freund–Rubin
background of (Euclidean) eleven-dimensional supergravity with a round metric
on the $\textrm{S}^{7}$ Freund:1980xh , and is dual to the superconformal ABJM
theory Aharony:2008ug  placed on S3. This vacuum sits at the origin of field
space, namely, ${z_{i}=\tilde{z}_{i}=0}$, and all our solutions will reach it
at $|\mu|\rightarrow\infty$ with a fall-off for the scalars of the form

$\begin{array}[]{rcll}z_{i}&=&a_{i}\,e^{-g|\mu|}+b_{i}\,e^{-2g|\mu|}+\ldots&,\\\\[5.69054pt]
\tilde{z}_{i}&=&\tilde{a}_{i}\,e^{-g|\mu|}+\tilde{b}_{i}\,e^{-2g|\mu|}+\ldots&.\end{array}$
(8)

According to the AdS/CFT correspondence, the fall-off coefficients
$(a_{i},b_{i})$ and $(\tilde{a}_{i},\tilde{b}_{i})$ in the expansions (8) are
related to sources and VEV’s in the field theory dual to the supergravity
solution. However, the precise identification turns out to be very subtle as
the combination $z_{i}-\tilde{z}_{i}$ should be quantised using regular
boundary conditions whereas alternate boundary conditions should be used for
$z_{i}+\tilde{z}_{i}$. A careful analysis of the boundary limit of the bulk
supersymmetry performed in Freedman:2013oja  yielded the following dictionary.
Sources in the dual field theory are identified with the combinations

$a_{i}-\tilde{a}_{i}\hskip 8.53581pt\textrm{ and }\hskip
8.53581pt\left(b_{i}-\frac{\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}}{\tilde{a}_{i}}\right)+\left(\tilde{b}_{i}-\frac{a_{1}a_{2}a_{3}}{a_{i}}\right)\
,$ (9)

whereas VEV’s are given by

$a_{i}+\tilde{a}_{i}\hskip 8.53581pt\textrm{ and }\hskip
8.53581pt\left(b_{i}-\frac{\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}}{\tilde{a}_{i}}\right)-\left(\tilde{b}_{i}-\frac{a_{1}a_{2}a_{3}}{a_{i}}\right)\
.$ (10)

This identification of conjugate variables (sources and VEV’s) is achieved by
selecting a holographic renormalisation scheme compatible with supersymmetry.
As stated in Freedman:2013oja , the counterterms of holographic
renormalisation have a universal structure and must be valid for all solutions
of the classical field equations of a given bulk theory. We will then follow
the holographic renormalisation prescription in Freedman:2013oja  to compute
the on-shell action and gravitational free energy of the supergravity
solutions constructed in this work.

##  III Euclidean solutions

The simplest solution to the system of second order equations in (4) is given
by a pure EAdS4 vacuum of the form

$\textrm{EAdS${}_{4}\,$ :}\hskip 8.53581pte^{2A}=\sinh^{2}(g\mu)\hskip
11.38109pt\textrm{ and }\hskip 11.38109ptz_{i}=\tilde{z}_{i}=0\ .$ (11)

This solution is maximally supersymmetric within the full $\mathcal{N}=8$
supergravity of deWit:1982ig , so it satisfies the BPS equations in (7). As
mentioned before, it uplifts to the $\textrm{EAdS}_{4}\times\textrm{S}^{7}$
vacuum solution of (Euclidean) eleven-dimensional supergravity with a round
metric on the $\textrm{S}^{7}$ Freund:1980xh , and is dual to the
superconformal ABJM theory Aharony:2008ug  on S3.

###  III.1 Solutions with a non-backreacted geometry

Let us start by presenting a class of solutions with a non-backreacted EAdS4
geometry and non-trivial profiles for the scalar fields. This is possible in a
spacetime with Euclidean signature because $z_{i}$ and $\tilde{z}_{i}$ are
independent fields and only the products of $z_{i}$ and $\tilde{z}_{i}$ (and
also of their derivatives) couple to the metric $A$ function in the Einstein
equation (first equation in (4)). Therefore, non-backreacted solutions exist
provided $z_{i}\tilde{z}_{i}=0$ $\forall i$.

The simplest case involves just one non-zero scalar, $z_{i}$ or
$\tilde{z}_{i}$, and the rest vanishing. For concreteness, let us consider
$z_{1}$ or $\tilde{z}_{1}$ being non-zero and
$z_{2}=z_{3}=\tilde{z}_{2}=\tilde{z}_{3}=0$. Within this setup, there is a
singular solution given by

$e^{2A}=\sinh^{2}(g\mu)\hskip 5.69054pt,\hskip 5.69054ptz_{1}=0\hskip
5.69054pt,\hskip
5.69054pt\tilde{z}_{1}=\frac{\tilde{c}_{1}}{\sinh^{2}\left(\frac{1}{2}g\mu\right)}\
,$ (12)

and a regular solution given by

$e^{2A}=\sinh^{2}(g\mu)\hskip 5.69054pt,\hskip
5.69054ptz_{1}=\frac{c_{1}}{\cosh^{2}\left(\frac{1}{2}g\mu\right)}\hskip
5.69054pt,\hskip 5.69054pt\tilde{z}_{1}=0\ .$ (13)

These solutions satisfy the second order equations in (4) as well as the BPS
equations (7) for the upper choice of sign therein. Therefore, they are
supersymmetric solutions. Moreover, if we demand a non-backreacted EAdS4
geometry, these two solutions are the only BPS solutions of the system with
the two fields $z_{1}$ and $\tilde{z}_{1}$ and the upper sign choice in (7).

Some comments about the solutions (12)-(13) are in order. Firstly, the
exchange $z_{1}\leftrightarrow\tilde{z}_{1}$ in (12)-(13) is a symmetry of the
second order equations of motion (4) but not of the BPS equations (7): the
exchange $z_{1}\leftrightarrow\tilde{z}_{1}$ amounts to a different sign
choice in the BPS equations (7) so that the preserved supersymmetries are
constructed using either left-invariant or right-invariant Killing spinors on
the S3. Secondly, solutions involving just one pair of fields
$(z_{i},\tilde{z}_{i})$ like, for example, the pair $(z_{1},\tilde{z}_{1})$ in
the solutions above, have an enhancement of symmetry from the
$\textrm{U}(1)^{4}$ generic symmetry of the STU-model to an
$\textrm{SU}(2)^{4}\sim\textrm{SO}(4)\times\textrm{SO}(4)\subset\textrm{SO}(8)$
symmetry. This $\textrm{SO}(4)\times\textrm{SO}(4)$ symmetry becomes manifest
when the solutions are uplifted to backgrounds of 11D supergravity using the
formulae in the Appendix A.1. The internal geometry is of the form
$\textrm{S}^{7}=\mathcal{I}\times\textrm{S}^{3}_{1}\times\textrm{S}^{3}_{2}$
with the two three-spheres $\textrm{S}^{3}_{1,2}$ being responsible for the
residual $\textrm{SO}(4)\times\textrm{SO}(4)$ symmetry of the solutions.
Thirdly, from (58)-(61) and (62)-(66), it becomes clear that
$c_{1}\in\mathbb{R}$ renders the 11D metric real and the four-form flux purely
imaginary. Also, the unit-disk normalisation condition along the flows
requires $|z_{1}(0)|=|c_{1}|<1$.

Since the bulk Lagrangian in (1)-(2) does not mix the different scalars, the
above solutions can be generalised to the STU-model. However, once generalised
to the STU-model, they become solutions of the second order equations (4) but
no longer of the BPS equations (7). Therefore they turn into non-
supersymmetric solutions. The singular solution in (12) is straightforwardly
generalised to the STU-model as

$e^{2A}=\sinh^{2}(g\mu)\hskip 5.69054pt,\hskip 5.69054ptz_{i}=0\hskip
5.69054pt,\hskip
5.69054pt\tilde{z}_{i}=\frac{\tilde{c}_{i}}{\sinh^{2}\left(\frac{1}{2}g\mu\right)}\
,$ (14)

whereas the regular solution in (13) has a generalisation of the form

$e^{2A}=\sinh^{2}(g\mu)\hskip 5.69054pt,\hskip
5.69054ptz_{i}=\frac{c_{i}}{\cosh^{2}\left(\frac{1}{2}g\mu\right)}\hskip
5.69054pt,\hskip 5.69054pt\tilde{z}_{i}=0\ .$ (15)

We will further study the regular solution (15) and compute its gravitational
on-shell action and free energy as a function of the parameters
$c_{i}\in\mathbb{R}$. Note that the unit-disk normalisation condition along
the scalars flow implies $|z_{i}(0)|=|c_{i}|<1$ $\forall i$. The uplift of the
regular solution (15) to eleven-dimensional supergravity requires to
generalise the Appendix A.1 to the full STU-model as done in Azizi:2016noi .
This goes beyond the scope of this note.

###  III.2 Solutions with a backreacted geometry

Inspired by the Janus solutions of Anabalon:2022fti , let us minimally modify
the EAdS4 geometry in (11) and present two new classes of analytic and multi-
parametric Euclidean solutions of the STU-model. These two classes of
solutions are generically non-supersymmetric although, as we will see, a
supersymmetric limit can still be considered.

#### Soliton solutions

In the first class of solutions, the spacetime metric (3) is specified by a
function

$e^{2A}=\frac{\sinh^{2}(g\mu)}{k^{2}}\ ,$ (16)

with $k^{2}>0$ so that the metric is real and Euclidean. Whenever $k^{2}\neq
1$, the spacetime metric possesses a singularity at $\mu=0$, as can be seen
from the scalar curvature
${R=6g^{2}\left(k^{2}-\cosh(2g\mu)\right)/\sinh^{2}(g\mu)}$. This restricts
the range of the solution to $\mu\in\mathbb{R}^{+}$, so we will refer to these
solutions as solitons.

The metric function in (16) requires both $z_{i}$ and $\tilde{z}_{i}$ scalars
to flow simultaneously. The profiles for these fields are of the form

$z_{i}=\frac{\lambda_{i}\sqrt{k_{i}^{2}-1}}{k_{i}+\cosh(g\mu)}\hskip
14.22636pt,\hskip
14.22636pt\tilde{z}_{i}=\frac{\lambda_{i}^{-1}\sqrt{k_{i}^{2}-1}}{k_{i}-\cosh(g\mu)}\
,$ (17)

with arbitrary parameters $\lambda_{i}$ and $k_{i}$ subject to the condition

$\sum_{i=1}^{3}k_{i}^{2}=k^{2}+2\ .$ (18)

This class of solutions has a metric singularity at ${\mu=0}$ together with a
divergence of the scalars whenever ${k^{2}_{i}>1}$. However, such a scalar
divergence can be eliminated by taking $k_{i}^{2}\leq 1$ with the constraint
${\sum k_{i}^{2}>2}$ so that $k^{2}>0$ in (18). The unit-disk normalisation
condition then fixes $|\lambda_{i}|^{-2}=\frac{1-k_{i}}{1+k_{i}}$ for the
scalar flows to reach the boundary of the unit-disk at $\mu=0$, i.e.
$|z_{i}(0)|=|\tilde{z}_{i}(0)|=1$. These singular solutions are very much
alike the Lorentzian flows to Hades in Anabalon:2022fti .

Interestingly, when two out of the three $k^{2}_{i}$ in (17) are set to unity,
one is left with a singular but supersymmetric solution with
$\textrm{SO}(4)\times\textrm{SO}(4)\subset\textrm{SO}(8)$ residual symmetry.
For example, setting $k^{2}_{2}=k^{2}_{3}=1$ reduces (18) to $k_{1}^{2}=k^{2}$
and yields

$z_{1}=e^{i\beta}\frac{k+1}{k+\cosh(g\mu)}\hskip 5.69054pt,\hskip
5.69054pt\tilde{z}_{1}=e^{-i\beta}\frac{k-1}{k-\cosh(g\mu)}\ ,$ (19)

with arbitrary $\beta\in\mathbb{R}$ and $0<k^{2}<1$. The solution (19)
satisfies the BPS equations (7). Unfortunately, when uplifted to a background
of 11D supergravity using the formulae in the Appendix A.1, the singularity at
$\mu=0$ persists as it happened for the dual of the Coulomb branch flows
investigated in Anabalon:2022fti . More concretely, the 11D Ricci scalar goes
as $\hat{R}\propto\mu^{-4/3}$ around $\mu=0$. It would be interesting to
further investigate the nature of this supersymmetric singularity and its
potential implications.

#### Wormhole solutions

The second class of solutions has a regular spacetime metric (3) specified by
the metric function

$e^{2A}=\frac{\cosh^{2}(g\mu)}{k^{2}}\ ,$ (20)

with $k^{2}>0$ so that the metric is real and Euclidean, and
${\mu\in\mathbb{R}}$. The radius of the $\textrm{S}^{3}$ in the spacetime
geometry (3) reaches a minimum size $(gk)^{-1}$ at $\mu=0$ and so we will
refer to this second class of solutions as wormholes.

The metric function in (20) requires both $z_{i}$ and $\tilde{z}_{i}$ to flow
with profiles

$z_{i}=\frac{\lambda_{i}\sqrt{k_{i}^{2}+1}}{k_{i}-\sinh(g\mu)}\hskip
14.22636pt,\hskip
14.22636pt\tilde{z}_{i}=\frac{\lambda^{-1}_{i}\sqrt{k_{i}^{2}+1}}{k_{i}+\sinh(g\mu)}\
,$ (21)

with arbitrary parameters $\lambda_{i}$ and $k_{i}$ subject to

$\sum_{i=1}^{3}k_{i}^{2}=k^{2}-2\ .$ (22)

Importantly, when $0<k^{2}<2$, the condition (22) has solutions of the form
$k_{i}=i\,\mathbf{k}_{i}$ with ${\mathbf{k}_{i}\in\mathbb{R}}$. The scalars
are non-singular and necessarily complex in this case, and read

$z_{i}=\frac{\lambda_{i}\sqrt{1-\mathbf{k}_{i}^{2}}}{i\,\mathbf{k}_{i}-\sinh(g\mu)}\hskip
14.22636pt,\hskip
14.22636pt\tilde{z}_{i}=\frac{\lambda^{-1}_{i}\sqrt{1-\mathbf{k}_{i}^{2}}}{i\,\mathbf{k}_{i}+\sinh(g\mu)}\
.$ (23)

As a result, whenever the real vector
$\vec{\mathbf{k}}\equiv(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})$ (with
$\mathbf{k}_{i}\neq 0$ $\forall i$) specifies a point on a two-sphere of
radius $(2-k^{2})>0$, the corresponding solution is fully regular. However,
the unit-disk parameterisation of the scalars still requires
$|z_{i}|,|\tilde{z}_{i}|<1$ all along the scalar flow. Provided this
normalisation condition holds, the solution describes an Euclidean wormhole
within the STU-model in the $\textrm{SO}(8)$ gauged supergravity. Lastly, the
supersymmetric limit of setting two out of the three $\mathbf{k}_{i}^{2}$ to
unity in (23) is no longer compatible with the condition (22), which reduces
in this case to $-\mathbf{k}_{1}^{2}=k^{2}$.

Example: A simple class of regular Euclidean wormholes follows from the
identification $\mathbf{k}_{i}^{2}=(2-k^{2})/3$ $\forall i$. In this case one
has

$|\lambda_{i}|^{-2}|z_{i}(\mu)|^{2}=|\lambda_{i}|^{2}|\tilde{z}_{i}(\mu)|^{2}=\frac{2(1+k^{2})}{(1-2k^{2})+3\cosh(2g\mu)}\
.$ (24)

Since $0<k^{2}<2$, the right hand side of (24) is regular and has a maximum at
$\mu=0$. Then the unit-disk normalisation condition requires

$\begin{array}[]{rcrcll}|z_{i}(0)|^{2}&=&|\lambda_{i}|^{2}\left(\dfrac{3}{2-k^{2}}-1\right)&<&1&,\\\\[11.38109pt]
|\tilde{z}_{i}(0)|^{2}&=&|\lambda_{i}|^{-2}\left(\dfrac{3}{2-k^{2}}-1\right)&<&1&.\end{array}$
(25)

Setting for simplicity $\lambda_{i}=e^{i\beta}$ $\forall i$, with arbitrary
$\beta\in\mathbb{R}$, the scalars take the final form

$z_{i}=-\tilde{z}_{i}^{*}=e^{i\beta}\frac{\sqrt{1+k^{2}}}{i\sqrt{2-k^{2}}-\sqrt{3}\sinh(g\mu)}\
,$ (26)

are regular and satisfy $\tilde{z}_{i}=-z_{i}^{*}\in\mathbb{C}$. Moreover,
they are compatible with the unit-disk parameterisation condition whenever
$0<k^{2}<\frac{1}{2}$. Note also that both the dilatonic $z_{i}+\tilde{z}_{i}$
and axionic $z_{i}-\tilde{z}_{i}$ combinations run non-trivially in this
example.

Finally, it is also worth mentioning that solutions with $z_{1}=z_{2}=z_{3}$
(same for $\tilde{z}_{i}$), like the one in (26), feature an enhancement of
symmetry from the generic $\textrm{U}(1)^{4}$ symmetry of the STU-model to an
$\textrm{SU}(3)\times\textrm{U}(1)^{2}\subset\textrm{SO}(8)$ symmetry.
However, supersymmetry is not preserved in the bulk as the BPS equations (7)
are not satisfied. These solutions are straightforwardly uplifted to Euclidean
solutions of eleven-dimensional supergravity using the uplift formulae in the
Appendix A.2: the result is a complex metric and a complex (not purely
imaginary) four-form flux in 11D.

The complex nature of Euclidean solutions should not be surprising. Metrics
with real and imaginary parts are already encountered in the Wick rotation of
spinning black holes. Also the Wick rotation of the electrically charged four-
dimensional Reissner-Nordström-AdS black hole has a purely imaginary gauge
field that yields a complex metric when uplifted to eleven dimensions. Lastly,
black rings do not even have everywhere regular real Euclidean sections
Elvang:2006dd .

##  IV On-shell action and gravitational free energy

The evaluation of the gravitational on-shell action gives rise to divergences
due to the infinite volume in the integral of the bulk action (1). Such
divergences are cured by applying the by now standard procedure of holographic
renormalisation Bianchi:2001kw . However, the holographic renormalisation
procedure comes along with an ambiguity: local finite counter-terms can be
added to the renormalised action. This reflects the scheme dependence of the
renormalisation procedure in quantum field theory. As concluded in
Freedman:2013oja  after the study of the projection of the bulk supersymmetry
into the boundary, the identification of sources and VEV’s in (9)-(10)
requires to include specific cubic counter-terms in the holographic
renormalisation procedure. The inclusion of such cubic counter-terms was shown
to be crucial to have a matching between the gravitational free energy of the
supergravity flows constructed in Freedman:2013oja  and the free energy of the
dual ABJM theory on $\textrm{S}^{3}$ deformed with supersymmetric real mass
terms computed using localisation methods.

The multi-parametric families of Euclidean solutions we have constructed in
the previous section belong to the same bulk theory as the solutions
constructed in Freedman:2013oja . Moreover, the solutions here possess a
supersymmetric limit whenever two of the scalars $z_{i}$ (same for
$\tilde{z}_{i}$) can be smoothly set to vanish upon tuning of the free
parameters. This makes us follow the same renormalisation prescription of
Freedman:2013oja  and include the same (finite) cubic counter-terms when
renormalising the on-shell action. The result is an on-shell action of the
form

$2\kappa^{2}S_{\textrm{on-
shell}}=2\kappa^{2}S_{\textrm{bulk}}+S_{\textrm{GH}}+S_{a}+S_{z}\ ,$ (27)

with $S_{\textrm{bulk}}$ given in (1), that contains a set of boundary terms

$\begin{array}[]{rcll}S_{\textrm{GH}}&=&-\displaystyle\int_{\partial
M}d^{3}x\sqrt{h}K\ ,\\\\[11.38109pt]
S_{a}&=&\dfrac{1}{2g}\displaystyle\int_{\partial M}d^{3}x\sqrt{h}R(h)\
,\\\\[11.38109pt] S_{z}&=&2\,\displaystyle\int_{\partial
M}d^{3}x\,\sqrt{h}\,\left[e^{\mathcal{K}}\,W\,\tilde{W}\right]^{\frac{1}{2}}\\\\[8.53581pt]
&=&g\displaystyle\int_{\partial
M}d^{3}x\,\sqrt{h}\,\left(2+\displaystyle\sum_{i}z_{i}\tilde{z}_{i}\right.\\\
&&\left.\hskip
76.82243pt+\displaystyle\prod_{i}z_{i}+\displaystyle\prod_{i}\tilde{z}_{i}+\ldots\right)\
.\end{array}$ (28)

The above boundary terms include the standard counter-terms that follow from
the near-boundary analysis of a generic supergravity solution (see e.g.
Bianchi:2001kw ), the specific cubic counter-term discussed in
Freedman:2013oja  (i.e. last line of (28)), and additional terms – denoted by
the ellipsis – which vanish at the boundary when taking the limit
$|\mu|\rightarrow\infty$. The quantity $K$ entering $S_{\textrm{GH}}$ is the
trace of the extrinsic curvature defined as
$K_{ab}=\nabla_{(a}N_{b)}=\frac{1}{2}\mathcal{L}_{N}h_{ab}=\pm\frac{1}{2}h_{ab}^{\prime}$.
Here $N_{a}=\pm\delta_{a}^{\mu}$ is the vector normal to the boundary
$\partial M$ and $h_{ab}=g_{ab}-N_{a}N_{b}$ is the induced metric 111The
ambiguity of the sign in $N_{a}$ is because, in the wormhole solution, the
boundary consists of two disconnected pieces located at $\mu=\pm\infty$.. The
quantity $R(h)$ entering $S_{a}$ is the Ricci scalar of this metric. Finally,
the functions $\mathcal{K}$, $W$ and $\tilde{W}$ are the Kähler potential and
superpotentials given in (5) and (6). We note in passing that the cubic
counter-terms $z_{1}z_{2}z_{3}$ and $\tilde{z}_{1}\tilde{z}_{2}\tilde{z}_{3}$
in the last line of (28) break the $\textrm{SO}(1,1)_{\xi_{i}}$ symmetries of
the bulk action (1).

Starting from the on-shell action (27), the computation of the gravitational
free energy, $J_{\textrm{on-shell}}$, requires a precise identification of
sources and VEVs from the scalar expansions (8) around the boundary. The
reason why is that a Legendre transformation of $S_{\textrm{on-shell}}$ might
be needed for $J_{\textrm{on-shell}}$ to depend on the particular combinations
of leading and sub-leading coefficients in (8) specifying the sources in the
dual field theory. With the identification of sources given in (9), the
gravitational free energy takes the form

$J_{\textrm{on-shell}}=S_{\textrm{on-shell}}+\frac{1}{2\kappa^{2}}\Delta S\ ,$
(29)

with

$\Delta
S=-\frac{1}{2}\displaystyle\sum_{i}\int_{\textrm{S}^{3}}dx^{3}(a_{i}+\tilde{a}_{i})\left(\frac{\delta
S_{\textrm{on-shell}}}{\delta a_{i}}+\frac{\delta S_{\textrm{on-
shell}}}{\delta\tilde{a}_{i}}\right)\ .$ (30)

And it is precisely at this point where the finite cubic counter-terms
entering $S_{z}$ in (28) play a crucial role as they guarantee that
$\delta_{a_{i}}S_{\textrm{on-shell}}+\delta_{\tilde{a}_{i}}S_{\textrm{on-
shell}}\propto{(\tilde{b}_{i}-a_{1}a_{2}a_{3}/a_{i})+(b_{i}-\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}/\tilde{a}_{i})}$
and, therefore, the gravitational free energy $J_{\textrm{on-shell}}$ is a
function of the sources in (9).

###  IV.1 Non-backreacted EAdS4 solutions

Let us recall the non-backreacted regular solution in (15), namely,

$e^{2A}=\sinh^{2}(g\mu)\hskip 5.69054pt,\hskip
5.69054ptz_{i}=\frac{c_{i}}{\cosh^{2}\left(\frac{1}{2}g\mu\right)}\hskip
5.69054pt,\hskip 5.69054pt\tilde{z}_{i}=0\ .$ (31)

When reaching the boundary at ${\mu=\infty}$, the scalar fields feature the
asymptotic expansions

$z_{i}=-4c_{i}\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\,n\,e^{-ng\mu}\hskip
14.22636pt\textrm{ and }\hskip 14.22636pt\tilde{z}_{i}=0\ .$ (32)

A direct comparison between (8)-(10) and (32) shows that

$\begin{array}[]{rcl}a_{i}\mp\tilde{a}_{i}&=&4c_{i}\ ,\\\\[5.69054pt]
\left(b_{i}-\frac{\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}}{\tilde{a}_{i}}\right)\pm\left(\tilde{b}_{i}-\frac{a_{1}a_{2}a_{3}}{a_{i}}\right)&=&-16\left(\frac{c_{i}}{2}\pm\frac{c_{1}c_{2}c_{3}}{c_{i}}\right)\
,\end{array}$ (33)

and thus the sources in (9) and VEVs in (10) are generically activated if
$c_{i}\neq 0$. The evaluation of the on-shell action (27) on the solution (31)
gives

$S_{\textrm{on-shell}}\Big{|}_{\textrm{non-
back}}=S^{\textrm{EAdS}_{4}}_{\textrm{on-
shell}}\left(1+4\prod_{i}c_{i}\right)\ ,$ (34)

where

$S^{\textrm{EAdS}_{4}}_{\textrm{on-shell}}=\frac{2\pi^{2}}{\kappa^{2}g^{2}}\
,$ (35)

is the on-shell action of the EAdS4 solution in (11). The first contribution
to the r.h.s of (34) comes from the bulk term in (27) whereas the second
$c_{i}$-dependent contribution originates from the non-zero cubic counter-
terms in $S_{z}$ of (28). Without the cubic counter-terms in the holographic
renormalisation procedure, the on-shell action of the solution (31) would be
the same as that of EAdS4. Also, from (34), it follows that the on-shell
action of the non-backreacted solution (31) is the same as that of EAdS4
whenever (at least) one of the scalars $z_{i}$ vanishes.

The gravitational free energy of the solution (31) is given by (29) and (30)
with

$\begin{array}[]{rcl}\dfrac{\delta S_{\textrm{on-shell}}}{\delta
a_{i}}&=&-\dfrac{\sqrt{g_{3}}}{16g^{2}\kappa^{2}}\left(\tilde{b}_{i}-\dfrac{a_{1}a_{2}a_{3}}{a_{i}}\right)\
,\\\\[11.38109pt] \dfrac{\delta S_{\textrm{on-
shell}}}{\delta\tilde{a}_{i}}&=&-\dfrac{\sqrt{g_{3}}}{16g^{2}\kappa^{2}}\left(b_{i}-\dfrac{\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}}{\tilde{a}_{i}}\right)\
,\end{array}$ (36)

where $g_{3}$ is the determinant of the round metric on the three-sphere
$\textrm{S}^{3}$ of unit radius. Substituting the $a_{i}$, $\tilde{a}_{i}$,
$b_{i}$ and $\tilde{b}_{i}$ coefficients that follow from direct comparison
between the general expansions in (8)-(10) and the ones in (32), the resulting
gravitational free energy is given by

$J_{\textrm{on-shell}}\Big{|}_{\textrm{non-
back}}=J^{\textrm{EAdS}_{4}}_{\textrm{on-
shell}}\left(1-\sum_{i}\mathcal{C}_{i}\right)\ ,$ (37)

where

$\mathcal{C}_{i}=c_{i}^{2}+\tfrac{2}{3}\,c_{1}c_{2}c_{3}\ ,$ (38)

and

$\begin{array}[]{rcl}J^{\textrm{EAdS}_{4}}_{\textrm{on-
shell}}&=&\dfrac{2\pi^{2}}{\kappa^{2}g^{2}}\ ,\end{array}$ (39)

is the universal gravitational free energy of the EAdS4 vacuum in (11). As a
consequence of F-extremisation, the gravitational free energy in (37) is
maximised at the conformal case, i.e. $c_{i}=0$ $\forall i$, so that

$J_{\textrm{on-shell}}\Big{|}_{\textrm{non-
back}}<J^{\textrm{EAdS}_{4}}_{\textrm{on-shell}}\ ,$ (40)

provided the unit-disk condition $|c_{i}|<1$ holds.

Let us comment on the gravitational free energy we have obtained. For generic
values of the parameters $c_{i}$, the solution (31) is non-supersymmetric and,
in principle, one could consider a renormalisation scheme different from the
supersymmetric one. For example, if applying the minimal subtraction scheme
that follows from the near-boundary analysis – so no finite cubic counter-
terms are included – the gravitational free energy changes to
$J^{\textrm{EAdS}_{4}}_{\textrm{on-shell}}\left(1-\sum_{i}c_{i}^{2}\right)$
with no mixing between the $z_{i}$’s. Positivity of this free energy further
restricts the range of the parameters $c_{i}$ to lie inside the unit radius
$c_{i}$-ball defined by the condition $\sum_{i}c_{i}^{2}<1$. In this note, and
in order to make contact with Freedman:2013oja  in the supersymmetric case, we
have applied the supersymmetric scheme and, therefore, the gravitational free
energy in (37) depends on the finite cubic counter-terms in $S_{z}$ of (28):
the cubic term in the r.h.s of (38) stems from such finite counter-terms.
Positivity of (37) then requires the parameters $c_{i}$ to lie inside a
deformed $c_{i}$-ball with four lumps defined by the cubic condition
$\sum_{i}c_{i}^{2}+2\,c_{1}c_{2}c_{3}<1$.

#### Supersymmetric limit and holography

Setting two $c_{i}$ to zero in the solution (31), i.e. ${c_{2}=c_{3}=0}$,
reduces it to the supersymmetric and
$\textrm{SO}(4)\times\textrm{SO}(4)\sim\textrm{SU}(2)^{4}$ invariant solution
in (13), namely,

$e^{2A}=\sinh^{2}(g\mu)\hskip 5.69054pt,\hskip
5.69054ptz_{1}=\frac{c_{1}}{\cosh^{2}\left(\frac{1}{2}g\mu\right)}\hskip
5.69054pt,\hskip 5.69054pt\tilde{z}_{1}=0\ .$ (41)

The general gravitational free energy in (37) reduces in this case to

$J_{\textrm{on-
shell}}\Big{|}_{\textrm{SO}(4)\times\textrm{SO}(4)}=J^{\textrm{EAdS}_{4}}_{\textrm{on-
shell}}\left(1-c_{1}^{2}\right)\ .$ (42)

The solution in (41) lies precisely at the intersection between the general
class of non-supersymmetric solutions in (31) and the class of
${\mathcal{N}=2}$ flows put forward in Freedman:2013oja  describing the
holography of $F$-maximisation Jafferis:2010un .

Following Freedman:2013oja , let us look at the field theory dual of the
solution (41). This is the ABJM superconformal field theory (SCFT) of
Aharony:2008ug  placed on (a unit radius) S3 and deformed with a specific
supersymmetric real mass parameter $\delta_{1}$. In general, there are three
such real mass parameters $\delta_{i}$ ($i=1,2,3$) compatible with
$\mathcal{N}=2$ supersymmetry which modify the assignment of
$\textrm{U}(1)_{\textrm{R}}$ R-charges. The real mass parameters $\delta_{i}$
break conformality and also the
$\textrm{U}(1)_{\textrm{R}}\times\textrm{SU}(2)\times\textrm{SU}(2)\times\textrm{U}(1)_{b}\subset\textrm{SO}(8)_{\textrm{R}}$
symmetry manifest in the $\mathcal{N}=2$ superfield formulation of the
(undeformed) ABJM SCFT down to its Cartan subgroup
$\textrm{U}(1)_{\textrm{R}}\times\textrm{U}(1)\times\textrm{U}(1)\times\textrm{U}(1)_{b}\subset\textrm{SO}(8)_{\textrm{R}}$.

In the $\mathcal{N}=2$ formulation of Aharony:2008ug , ABJM theory is a SCFT
with gauge group $\textrm{U}(N)\times\textrm{U}(N)$ and Chern–Simons (CS)
levels $(k,-k)$ – in our case $k=1$ and the internal space in the gravity side
is the round S7 –. The theory consists of two vectors multiplets
$(A,\sigma,\lambda,D)$ and
$(\tilde{A},\tilde{\sigma},\tilde{\lambda},\tilde{D})$ for the
$\textrm{U}(N)\times\textrm{U}(N)$ gauge group together with four chiral
matter multiplets $(Z^{a},\chi^{a},F^{a})$ and $(W_{a},\eta_{a},G_{a})$, with
$a=1,2$, transforming in the $(\bf{\bar{N}},\bf{N})$ and
$(\bf{N},\bf{\bar{N}})$ representations of the gauge group. There is also a
quartic superpotential for the matter multiplets which has R-charge $2$ and is
invariant under the
$\textrm{SU}(2)\times\textrm{SU}(2)\times\textrm{U}(1)_{b}$ flavour symmetry.
It is given by

$W\propto\textrm{Tr}\left(\epsilon_{ab}\epsilon^{cd}Z^{a}W_{c}Z^{b}W_{d}\right)\
.$ (43)

Particularising the Lagrangian in Freedman:2013oja  to describe,
holographically, the solution in (41) one finds

$\mathcal{L}=\mathcal{L}_{\textrm{SCFT}}+\delta_{1}\left[\mathcal{O}^{1}_{B}-\delta_{1}\mathcal{O}_{S}\right]+\delta_{1}\mathcal{O}^{1}_{F}\
,$ (44)

where $\mathcal{L}_{\textrm{SCFT}}$ is the ABJM SCFT Lagrangian and
$\delta_{1}$ is a specific supersymmetric real mass parameter that turns on
bosonic and fermionic operators. In particular, it turns on the bosonic
$\mathcal{O}^{1}_{B}$ and fermionic $\mathcal{O}^{1}_{F}$ operators

$\begin{array}[]{rcl}\mathcal{O}^{1}_{B}&=&\textrm{tr}\left[Z^{\dagger}_{a}Z^{a}-W^{\dagger
a}W_{a}\right]\ ,\\\\[11.38109pt]
\mathcal{O}^{1}_{F}&=&\textrm{tr}\left[\chi^{\dagger
a}\chi_{a}-\eta^{\dagger}_{a}\eta^{a}\right]+2i(\sigma-\tilde{\sigma})\mathcal{O}_{S}\
,\end{array}$ (45)

dual to the proper-scalar $z_{1}+\tilde{z}_{1}$ and the pseudo-scalar
$z_{1}-\tilde{z}_{1}$ in the $\textrm{SO}(8)$ supergravity, together with an
additional Konishi-like bosonic operator

$\mathcal{O}_{S}=\textrm{tr}\left[Z^{\dagger}_{a}Z^{a}+W^{\dagger
a}W_{a}\right]\ ,$ (46)

which does not have an associated scalar in the $\textrm{SO}(8)$ supergravity
222Spinless fields in the $\textrm{SO}(8)$ supergravity transform in the
$\bf{35}_{v}$ (proper-scalars) and $\bf{35}_{c}$ (pseudo-scalars) symmetric
and traceless representations of $\textrm{SO}(8)$. The bosonic operator
$\mathcal{O}_{S}$ in (46) corresponds to the trace of the boson bilinears and,
therefore, has no associated scalar in the $\textrm{SO}(8)$ supergravity..
Remarkably, amongst the operators with a dual spinless field in the
$\textrm{SO}(8)$ supergravity, those in (45) are the only two singlets under
the $\textrm{SU}(2)\times\textrm{SU}(2)\times\textrm{U}(1)_{b}$ flavour
symmetry of the ABJM SCFT in the $\mathcal{N}=2$ formulation: under this
symmetry the chiral multiplets $(Z^{a},\chi^{a},F^{a})$ and
$(W_{a},\eta_{a},G_{a})$ transform in the ${(\bf{2},\bf{1})}_{1}$ and
${(\bf{1},\bf{\bar{2}})}_{-1}$ representations, respectively 333See Table 1 of
Freedman:2013oja  for a summary of the
$\textrm{U}(1)_{\textrm{R}}\times\textrm{SU}(2)\times\textrm{SU}(2)\times\textrm{U}(1)_{b}$
charges of the various fields in the ABJM SCFT.

The parameter $\delta_{1}$ in (44) determines the assignment of
$\textrm{U}(1)_{\textrm{R}}$ R-charges for the scalar component of the chiral
superfields to be

$R[Z^{1}]=R[Z^{2}]=\tfrac{1}{2}+\delta_{1}\,,\,R[W_{1}]=R[W_{2}]=\tfrac{1}{2}-\delta_{1}\
.$ (47)

On the other hand, the free energy of the ABJM theory on $\textrm{S}^{3}$
deformed with $\mathcal{N}=2$ supersymmetric real mass terms can be obtained
as a function of a set of trial R-charges using localisation methods
Kapustin:2009kz ; Jafferis:2010un . To leading order in $N$ and for CS level
$k=1$ the result is Jafferis:2011zi

$\mathcal{F}=\mathcal{F}_{0}\sqrt{16\,R[Z^{1}]\,R[Z^{2}]\,R[W_{1}]\,R[W_{2}]}\
,$ (48)

with $\mathcal{F}_{0}=\sqrt{2}\pi N^{3/2}/3$ and where the $R$’s are the trial
charges of the four chiral matter fields in the theory. The superpotential of
the undeformed ABJM SCFT theory (43) is quartic on the matter fields and has
R-charge $2$. This implies that $R[Z^{1}]+R[Z^{2}]+R[W_{1}]+R[W_{2}]=2$, rule
that is obeyed by the R-charges in (47). A direct substitution of the
R-charges (47) into (48) yields

$\mathcal{F}_{\textrm{SO}(4)\times\textrm{SO}(4)}=\mathcal{F}_{0}\,(1-(2\delta_{1})^{2})\
,$ (49)

which matches the gravitational free energy in (42) for the supersymmetric
$\textrm{SO}(4)\times\textrm{SO}(4)$ invariant solution (41) provided

$\mathcal{F}_{0}=J^{\textrm{EAdS}_{4}}_{\textrm{on-shell}}\ ,$ (50)

and the identification ${c_{1}=2\delta_{1}}$. The extremisation of the free
energy in (49) fixes $\delta_{1}=0$ and assigns canonical R-charges of
$\frac{1}{2}$ to the four chiral fields in (47). The free energy in (49) then
reduces to $\mathcal{F}_{0}$ which, using the holographic mapping between
$\kappa^{2}$ and $N$ in the ABJM SCFT at CS level $k=1$, matches the
gravitational free energy of the EAdS4 vacuum in (39).

###  IV.2 Backreacted wormhole solutions

Let us now investigate the backreacted solution in (20) and (23). As we
demonstrated with the example of Section III.2, the various parameters in the
solution can be chosen such that the solution is regular in the domain
$\mu\in(-\infty,\infty)$, and the scalars stay within the unit-disk all along
the flow. Having one of these regular and well-defined solutions in mind, we
will proceed an compute its on-shell action.

The boundary of spacetime now consists of two pieces located at
$\mu=\pm\infty$. The asymptotic expansions of the scalars in (23) around
$\mu=\pm\infty$ yield

$\begin{array}[]{lll}z_{i}&=&2\,\lambda_{i}\,\sqrt{1-\mathbf{k}^{2}_{i}}\,\big{(}\pm
e^{-g|\mu|}-2\,i\,\mathbf{k}_{i}\,e^{-2g|\mu|}+\ldots\big{)}\
,\\\\[11.38109pt]
\tilde{z}_{i}&=&2\,\lambda_{i}^{-1}\,\sqrt{1-\mathbf{k}^{2}_{i}}\,\big{(}\mp
e^{-g|\mu|}-2\,i\,\mathbf{k}_{i}\,e^{-2g|\mu|}+\ldots\big{)}\ ,\end{array}$
(51)

where the upper (lower) sign corresponds to the piece of the boundary at
$\mu=+\infty$ ($\mu=-\infty$). Using the asymptotic expansions in (51) to
evaluate the boundary terms, the on-shell action (27) for the wormholes takes
the form

$S_{\textrm{on-
shell}}\Big{|}_{\textrm{WH}}=S_{\textrm{cubic}}\Big{|}_{\partial
M_{+}}+S_{\textrm{cubic}}\Big{|}_{\partial M_{-}}=0\ .$ (52)

The finite contributions coming from the cubic counter-terms in $S_{z}$ of
(28) at the two pieces $\partial M_{\pm}$ (respectively at $\mu=\pm\infty$) of
the boundary read

$S_{\textrm{cubic}}\Big{|}_{\partial
M_{\pm}}=\mp\frac{2\pi^{2}}{g^{2}k^{3}}\left(\Lambda-\frac{1}{\Lambda}\right)\displaystyle\prod_{i}\sqrt{1-\mathbf{k}_{i}^{2}}\
,$ (53)

with $\Lambda\equiv\prod_{j}\lambda_{j}$. Except for the finite contributions
in (53), all the other contributions to the on-shell action (27) cancel
separately on each piece of the boundary of the wormhole.

Using the boundary expansions for the scalar fields in (51), one finds that

$\begin{array}[]{rcl}\dfrac{\delta S_{\textrm{on-shell}}}{\delta
a_{i}}&=&-\dfrac{\sqrt{g_{3}}}{16g^{2}\kappa^{2}k^{3}}\left(\tilde{b}_{i}-\dfrac{a_{1}a_{2}a_{3}}{a_{i}}\right)\
,\\\\[11.38109pt] \dfrac{\delta S_{\textrm{on-
shell}}}{\delta\tilde{a}_{i}}&=&-\dfrac{\sqrt{g_{3}}}{16g^{2}\kappa^{2}k^{3}}\left(b_{i}-\dfrac{\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}}{\tilde{a}_{i}}\right)\
,\end{array}$ (54)

and an explicit computation of (30) for the wormholes in (20) and (23) yields
$\Delta S|_{\partial M_{+}}=-\Delta S|_{\partial M_{-}}$. As a result, the
total gravitational free energy vanishes again due to an exact cancellation
between the two pieces of the wormhole boundary. Namely,

$J_{\textrm{on-shell}}\Big{|}_{\textrm{WH}}=J_{\textrm{on-
shell}}\Big{|}_{\partial M_{+}}+J_{\textrm{on-shell}}\Big{|}_{\partial
M_{-}}=0\ .$ (55)

Importantly, this result is independent of the holographic renormalisation
scheme, as any other choice of cubic counter-terms would have also resulted in
a null contribution to the gravitational free energy – and also to the on-
shell action – due to the antisymmetry of the scalar boundary conditions (51)
at $\mu=\pm\infty$.

##  V Conclusions

In this note we have constructed new classes of $\textrm{U}(1)^{4}$-invariant
Euclidean solutions in the four-dimensional STU-model (with vectors turned
off) that arises upon compactification of 11D supergravity on S7. Together
with some potentially pathological examples featuring spacetime singularities
– like the solitons in Section III.2– we presented two classes of fully
regular multi-parametric Euclidean solutions.

The first class describes flows with running scalars $z_{i}$ ($i=1,2,3$) but a
non-backreacted EAdS4 spacetime. This is possible in Euclidean signature,
where the scalars $z_{i}$ and their would-be conjugates $\tilde{z}_{i}$ are
independent fields so that a non-trivial solution with $z_{i}\tilde{z}_{i}=0$
$\forall i$ has a vanishing energy-momentum tensor. We constructed the three-
parameter family of non-supersymmetric solutions in (15) and computed its on-
shell action and gravitational free energy employing holographic
renormalisation methods. For the particular choice of parameters $c_{i}\neq 0$
and $c_{j}=c_{k}=0$ in (15), with $i\neq j\neq k$, the solution features an
$\textrm{SO}(4)\times\textrm{SO}(4)$ symmetry enhancement, becomes
supersymmetric, and belongs to the class of flows put forward in
Freedman:2013oja . In this supersymmetric limit, the gravitational free energy
was shown to match that of ABJM placed on S3 and deformed with a specific real
mass term which, in the $\mathcal{N}=2$ formulation of the theory, preserves
the full $\textrm{SU}(2)\times\textrm{SU}(2)\times\textrm{U}(1)_{b}$ flavour
symmetry group. We also presented the embedding of this supersymmetric
solution into 11D supergravity and showed that, when the profile for the
running scalar is taken to be real, i.e. $c_{i}\in\mathbb{R}$, the uplift to
11D produces a real metric and a purely imaginary four-form flux. This agrees
with the standard statement that “axions” flip the sign of their kinetic term
in the Euclidean theory.

The second class describes regular wormholes with scalar profiles satisfying
$z_{i}\tilde{z}_{i}\neq 0$ and supporting the wormhole geometry. We
constructed the multi-parametric family of wormholes in (20) and (23), and
showed that a supersymmetric limit is not possible for these solutions. We
also computed the on-shell action and gravitational free energy of the
wormholes and found that the latter always vanishes irrespectively of
supersymmetry and the holographic renormalisation scheme. It would be
interesting to understand this result from a field-theoretic perspective.
However, the absence of supersymmetry makes a holographic test of the zero
gravitational free energy difficult.

Unlike for the non-backreacted flow solutions, the scalars $z_{i}$ and
$\tilde{z}_{i}$ are necessarily complex in the wormholes. In the particular
example with $\textrm{SU}(3)\times\textrm{U}(1)^{2}$ symmetry presented in
Section III.2, it happens that $\tilde{z}_{i}=-z_{i}^{*}\in\mathbb{C}$ (not
$\tilde{z}_{i}=z_{i}^{*}$), but each pair $(z_{i},\tilde{z}_{i})$ still
comprises two real degrees of freedom. In addition, the a priori axionic
combination $z_{i}-\tilde{z}_{i}$ has a non-trivial profile providing an
axionic charge to the wormhole. Upon uplift of this example to eleven
dimensions, both the 11D metric and the four-form flux turn out to have non-
trivial real and imaginary parts.

It is also interesting to contrast the class of wormholes presented in this
note with the Maldacena–Maoz construction of Euclidean wormholes
Maldacena:2004rf . In the latter, the foliation of EAdS space is additionally
compactified in order to generate the Euclidean wormhole. This
compactification procedure imposes identifications that break supersymmetry
even asymptotically. In our case, and despite being non-supersymmetric, the
wormholes asymptote supersymmetric EAdS vacua on both sides of the solution,
thus ensuring a theory with a stable ground state. It would be interesting to
further investigate the stability of the wormholes presented here also as part
of the swampland program, and to assess their allowability in light of the
recent discussion in Witten:2021nzp .

Finally, the class of wormholes presented in this note was inspired by (the
Euclidean continuation of) the Lorentzian Janus solutions put forward in
Anabalon:2022fti . Supersymmetric Lorentzian Janus solutions with
identifications that make them wormhole-like (e.g. with non-trivial first
homotopy group) have been constructed in Anabalon:2018rzq ; Anabalon:2020loe
using vector fields instead of scalars. Their generalisation to include non-
trivial running scalars should be possible and could serve as a starting point
to construct analytic families of supersymmetric Euclidean wormholes in
M-theory. We hope to come back to this and related issues in the future.

Acknowledgements: We are grateful to C. Hoyos, D. Rodríguez-Gómez and P. Soler
for interesting conversations. The work of AA is supported in part by the
FAPESP visiting researcher award 2022/11765-7 and the FONDECYT grants 1200986,
1210635, 1221504 and 1230853. The work of AG is supported by the Spanish
government grant PGC2018-096894-B-100 and by the Principado de Asturias
through the grant FC-GRUPIN-IDI/2018/000174.

##  Appendix A Uplifts to 11D supergravity

The bosonic sector of eleven-dimensional supergravity consists of a metric
field $\hat{G}_{MN}$ and a three-form potential $\hat{A}_{(3)}$ with field
strength $\hat{F}_{(4)}=d\hat{A}_{(3)}$. The Euclidean equations of motion
read Nakayama:2016xzs

$\begin{array}[c]{rll}d(*_{11}\hat{F}_{(4)})+\frac{i}{2}\,\hat{F}_{(4)}\wedge\hat{F}_{(4)}&=&0\
,\\\\[5.69054pt]
\hat{R}_{MN}-\frac{1}{12}\left((\hat{F}\hat{F})_{MN}-\frac{1}{12}\,(\hat{F}\hat{F})\,\hat{G}_{MN}\right)&=&0\
,\end{array}$ (56)

where we have denoted
$(\hat{F}\hat{F})_{MN}\equiv\hat{F}_{MPQR}\,\hat{F}_{N}{}^{PQR}$ and
$(\hat{F}\hat{F})\equiv\hat{F}_{PQRS}\,\hat{F}^{PQRS}$. In addition, the field
strength $\hat{F}_{(4)}$ satisfies the source-less Bianchi identity

$d\hat{F}_{(4)}=0\ .$ (57)

In this appendix we present the necessary formulae to uplift the simplest
four-dimensional solutions discussed in the main text to Euclidean solutions
of eleven-dimensional supergravity. More concretely, we present the embedding
of the $\textrm{SO}(4)\times\textrm{SO}(4)$ and
$\textrm{SU}(3)\times\textrm{U}(1)^{2}$ invariant sectors of the maximal SO(8)
gauged supergravity into 11D supergravity.

###  A.1 $\textrm{SO}(4)\times\textrm{SO}(4)$ invariant sector

When only one pair of scalars $(z_{i},\tilde{z}_{i})$ is non-zero – we will
choose $(z,\tilde{z})\equiv(z_{1},\tilde{z}_{1})$ without loss of generality –
the corresponding solutions belong to the $\textrm{SO}(4)\times\textrm{SO}(4)$
invariant sector of the maximal $\textrm{SO}(8)$ supergravity. The uplift of
this sector to Lorentzian supergravity in eleven dimensions can be found in
Bobev:2013yra . Here we extend the uplift formulae therein to Euclidean
signature and adapt it to the unit-disk parameterisation of the scalar fields
$z$ and $\tilde{z}$ we are using in this work.

The 11D (warped) geometry is of the form $\mathcal{M}_{4}\times\textrm{S}^{7}$
with
$\textrm{S}^{7}=\mathcal{I}\times\textrm{S}^{3}_{1}\times\textrm{S}^{3}_{2}$.
The two three-spheres $\textrm{S}^{3}_{1,2}$ in the internal geometry are
responsible for the residual ${\textrm{SO}(4)\times\textrm{SO}(4)}$ symmetry
of the solutions. Denoting ${\alpha\in[0,\frac{\pi}{2}]}$ the coordinate along
the interval $\mathcal{I}$, the 11D metric is given by

$ds_{11}^{2}=\Delta^{2}\left[ds_{4}^{2}+\frac{4}{g^{2}}\left(d\alpha^{2}+\frac{\cos^{2}\alpha}{f_{1}}ds^{2}_{\textrm{S}^{3}_{1}}+\frac{\sin^{2}\alpha}{f_{2}}ds^{2}_{\textrm{S}^{3}_{2}}\right)\right],$
(58)

where the external spacetime metric $ds_{4}^{2}$ is given in (3) and

$ds^{2}_{\textrm{S}^{3}_{1}}=\tfrac{1}{4}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}\right)\hskip
5.69054pt,\hskip
5.69054ptds^{2}_{\textrm{S}^{3}_{2}}=\tfrac{1}{4}\left(\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2}+\hat{\sigma}_{3}^{2}\right)\
,$ (59)

denote the line elements on the two internal three-spheres. The warping factor
in (58) takes the form

$\Delta^{2}=f_{1}^{\frac{1}{3}}\,f_{2}^{\frac{1}{3}}\ ,$ (60)

in terms of two functions

$\begin{array}[]{rcll}f_{1}&=&\cos^{2}\alpha\,\dfrac{(1+z)(1+\tilde{z})}{1-z\,\tilde{z}}+\sin^{2}\alpha&,\\\\[11.38109pt]
f_{2}&=&\sin^{2}\alpha\,\dfrac{(1-z)(1-\tilde{z})}{1-z\,\tilde{z}}+\cos^{2}\alpha&.\end{array}$
(61)

The four-form field strength of 11D supergravity has both a spacetime and an
internal piece, namely,

$\hat{F}_{(4)}=\hat{F}^{\textrm{st}}_{(4)}+\hat{F}^{\textrm{tr}}_{(4)}\ .$
(62)

The spacetime part takes the form

$\hat{F}^{\textrm{st}}_{(4)}=g\,h\,\textrm{vol}_{4}+g^{-1}\sin(2\alpha)\,\tilde{h}^{(3)}\wedge
d\alpha\ ,$ (63)

in terms of a function $\,h\,$ and a three-form $\,\tilde{h}^{(3)}\,$ given by

$\begin{array}[]{rcl}h&=&i\,\dfrac{3-z\,\tilde{z}+(z+\tilde{z})\,\cos(2\alpha)}{(1-z\tilde{z})}\
,\\\\[11.38109pt]
\tilde{h}^{(3)}&=&i\,\dfrac{(1-z^{2})*_{4}d\tilde{z}+(1-\tilde{z}^{2})*_{4}dz}{(1-z\,\tilde{z})^{2}}\
.\end{array}$ (64)

In the above expression, $*_{4}$ denotes the Hodge dual with respect to the
external spacetime metric (3). The internal part is given by

$\hat{F}^{\textrm{tr}}_{(4)}=d\hat{A}^{\textrm{tr}}_{(3)}\hskip
8.53581pt\textrm{ with }\hskip
8.53581pt\hat{A}^{\textrm{tr}}_{(3)}=g^{-3}\,h_{1}\,\textrm{vol}_{1}+g^{-3}\,h_{2}\,\textrm{vol}_{2}\
,$ (65)

in terms of the functions

$h_{1}=i\,8\,\dfrac{\cos^{4}\alpha}{f_{1}}\,\dfrac{z-\tilde{z}}{1-z\tilde{z}}\hskip
8.53581pt,\hskip
8.53581pth_{2}=i\,8\,\dfrac{\sin^{4}\alpha}{f_{2}}\,\dfrac{z-\tilde{z}}{1-z\tilde{z}}\
,$ (66)

and the volume forms on the two three-spheres $\textrm{vol}_{1,2}$. Note that
$\hat{F}^{\textrm{tr}}_{(4)}$ is non-zero whenever the axionic combination
$z-\tilde{z}$ is non-zero.

Due to the consistency of the truncation, any solution in the
$\textrm{SO}(4)\times\textrm{SO}(4)$ invariant sector of the $\textrm{SO}(8)$
gauged supergravity is also a solution of the Euclidean equations of motion in
(56) and satisfies the Bianchi identity (57). Note also that, under the
exchange $z\rightarrow\tilde{z}$, the 11D geometry in (58) remains invariant
whereas the four-form flux in (62) changes as
$\hat{F}^{\textrm{st}}_{(4)}\rightarrow\hat{F}^{\textrm{st}}_{(4)}$ and
$\hat{F}^{\textrm{tr}}_{(4)}\rightarrow-\hat{F}^{\textrm{tr}}_{(4)}$.

###  A.2 $\textrm{SU}(3)\times\textrm{U}(1)^{2}$ invariant sector

When the three pairs $(z_{i},\tilde{z}_{i})$ of scalar fields in the STU-model
are identified, i.e. $(z,\tilde{z})\equiv(z_{i},\tilde{z}_{i})$ $\forall i$,
the corresponding solutions belong to the
$\textrm{SU}(3)\times\textrm{U}(1)^{2}$ invariant sector of the maximal
$\textrm{SO}(8)$ supergravity. The uplift of this sector to eleven-dimensional
supergravity has been worked out in Pilch:2015dwa  (see also Larios:2019kbw
whose conventions we closely follow) in the case of Lorentzian signature. As
before, we extend the uplift formulae therein to Euclidean signature and adapt
it to the unit-disk parameterisation of the scalar fields $z$ and $\tilde{z}$
we have adopted.

The eleven-dimensional (warped) geometry is of the form
$\mathcal{M}_{4}\times\textrm{S}^{7}$ with
$\textrm{S}^{7}=\mathcal{I}\times\mathbb{CP}_{2}\times\textrm{S}_{\tau}^{1}\times\textrm{S}_{\psi}^{1}$.
The $\mathbb{CP}_{2}$ and $\textrm{S}_{\tau}^{1}\times\textrm{S}_{\psi}^{1}$
factors in the internal geometry are responsible for the
$\textrm{SU}(3)\times\textrm{U}(1)^{2}$ residual symmetry of the four-
dimensional solutions. Denoting $\alpha\in[0,\frac{\pi}{2}]\,$ the coordinate
along the interval $\mathcal{I}$, the 11D metric takes the form

$\begin{array}[]{lll}d\hat{s}_{11}^{2}&=&\frac{1}{2}\,f_{1}\,ds_{4}^{2}+2\,g^{-2}\Big{[}f_{2}\,d\alpha^{2}\\\\[5.69054pt]
&+&\cos^{2}\alpha\,\big{(}\,f_{3}\,\,ds^{2}_{\mathbb{CP}_{2}}+\sin^{2}\alpha\,f_{4}\,(d\tau+\sigma)^{2}\,\big{)}\\\\[5.69054pt]
&+&f_{5}\,\big{(}d\psi+\cos^{2}\alpha\,f_{6}\,(d\tau+\sigma)\big{)}^{2}\Big{]}\
,\end{array}$ (67)

with $\tau,\psi\in[0,2\pi]$. We have denoted $\sigma$ the one-form on
$\mathbb{CP}_{2}$ such that $d\sigma=2\bm{J}$ with $\bm{J}$ being the Kähler
form on $\,\mathbb{CP}_{2}$. The metric in (67) is fully specified in terms of
six functions $f_{1\ldots 6}$ given by

$\begin{array}[]{c}f_{1}=(1-z\tilde{z})^{-1}L^{\frac{1}{3}}\,H^{\frac{2}{3}}\hskip
8.53581pt,\hskip 8.53581ptf_{2}=f_{3}^{-2}=H^{\frac{2}{3}}L^{-\frac{2}{3}}\
,\\\\[5.69054pt]
f_{4}=(1-z\tilde{z})^{2}L^{-\frac{2}{3}}\,H^{\frac{2}{3}}\,K^{-1}\hskip
8.53581pt,\hskip 8.53581ptf_{5}=H^{-\frac{4}{3}}\,K\,L^{-\frac{2}{3}}\
,\\\\[5.69054pt]
f_{6}=\Big{[}L\,H+(z-\tilde{z})^{2}\cos(2\alpha)\Big{]}\,K^{-1}\ ,\end{array}$
(68)

with

$\begin{array}[]{ccll}H&=&1+z\tilde{z}-(z+\tilde{z})\cos(2\alpha)&,\\\\[5.69054pt]
K&=&1+(z\tilde{z})^{2}-2\,z\tilde{z}\,\cos(4\alpha)&,\\\\[5.69054pt]
L&=&(1+z)(1+\tilde{z})&.\end{array}$ (69)

The four-form field strength of 11D supergravity consists of a spacetime and
an internal piece

$\hat{F}_{(4)}=\hat{F}^{\textrm{st}}_{(4)}+\hat{F}^{\textrm{tr}}_{(4)}\ .$
(70)

The spacetime part reads

$\hat{F}_{(4)}^{\textrm{st}}=g\,h_{1}\,\text{vol}_{4}+g^{-1}\,\sin(2\alpha)\,\,h^{(3)}_{2}\wedge
d\alpha\ ,$ (71)

in terms of a zero-form $h_{1}$ and a three-form $h_{2}^{(3)}$ given by

$\begin{array}[]{rcll}h_{1}&=&i\,\dfrac{3\,(1+z\tilde{z})+(z+\tilde{z})\,(1-2\cos(2\alpha))}{2\sqrt{2}\,(1-z\tilde{z})}&,\\\\[11.38109pt]
h_{2}^{(3)}&=&i\,\dfrac{(1-z^{2})*_{4}d\tilde{z}+(1-\tilde{z}^{2})*_{4}dz}{\sqrt{2}\,(1-z\tilde{z})^{2}}&.\end{array}$
(72)

Again, $*_{4}$ denotes the Hodge dual with respect to the external spacetime
metric (3). Lastly, the internal part is given by

$\begin{array}[]{lll}\hat{F}_{(4)}^{\textrm{tr}}&=&2\sqrt{2}\,g^{-3}\Big{[}\,\sin(2\alpha)\,h^{(1)}_{3}\wedge
d\alpha\wedge d\psi\wedge(d\tau+\sigma)\\\\[5.69054pt]
&+&\cos^{4}\alpha\,\,h^{(1)}_{4}\wedge(d\tau+\sigma)\wedge\bm{J}\\\\[5.69054pt]
&+&\cos^{2}\alpha\,\cos(2\alpha)\,\,h_{5}^{(1)}\wedge
d\psi\wedge\bm{J}\\\\[5.69054pt]
&+&\cos^{2}\alpha\sin(2\alpha)\,h_{6}\,d\alpha\wedge
d\psi\wedge\bm{J}+\cos^{4}\alpha\,\,h_{7}\,\bm{J}\wedge\bm{J}\\\\[5.69054pt]
&+&\cos^{2}\alpha\,\sin(2\alpha)\,\,h_{8}\,d\alpha\wedge(d\tau+\sigma)\wedge\bm{J}\,\Big{]}\
,\end{array}$ (73)

in terms of the one-forms

$\begin{array}[]{lll}h^{(1)}_{3}&=&\dfrac{i}{2}\left(\dfrac{d\tilde{z}}{(1+\tilde{z})^{2}}-\dfrac{dz}{(1+z)^{2}}\right)\
,\\\\[11.38109pt]
h^{(1)}_{4}&=&h^{(1)}_{5}\,\,=\,\dfrac{i}{2H^{2}}\,\Big{[}\,\,\left(1-2\cos(2\alpha)\,\tilde{z}+\tilde{z}^{2}\right)\,dz\\\
&&\hskip
56.9055pt-\left(1-2\cos(2\alpha)\,z+z^{2}\right)\,d\tilde{z}\,\,\Big{]}\
,\end{array}$ (74)

and the zero-forms

$\begin{array}[]{lll}h_{6}&=&i\,2\,(1+z\tilde{z})\,\dfrac{(\tilde{z}-z)}{LH^{2}}\,\Big{(}1+z\tilde{z}+(z+\tilde{z})\sin^{2}\alpha\Big{)}\
,\\\\[11.38109pt] h_{7}&=&-i\,\dfrac{(\tilde{z}-z)}{2H}\ ,\\\\[11.38109pt]
h_{8}&=&i\,\dfrac{(\tilde{z}-z)}{H^{2}}\,\Big{(}1+z\tilde{z}+(z+\tilde{z})\sin^{2}\alpha\Big{)}\
.\end{array}$ (75)

Note that the purely internal part of $\hat{F}^{\textrm{tr}}_{(4)}$ is non-
zero whenever the axionic combination $z-\tilde{z}$ is non-zero.

By virtue of the consistency of the truncation, any solution in the
$\textrm{SU}(3)\times\textrm{U}(1)^{2}$ invariant sector of the
$\textrm{SO}(8)$ gauged supergravity is also a solution of the Euclidean
equations of motion in (56) and satisfies the Bianchi identity (57).

## References

  * (1) G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15 (1977) 2752–2756.
  * (2) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200.
  * (3) A. B. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett. 43 (1986) 730–732.
  * (4) D. L. Jafferis, I. R. Klebanov, S. S. Pufu, and B. R. Safdi, “Towards the F-Theorem: N=2 Field Theories on the Three-Sphere,” JHEP 06 (2011) 102, arXiv:1103.1181 [hep-th].
  * (5) I. R. Klebanov, S. S. Pufu, and B. R. Safdi, “F-Theorem without Supersymmetry,” JHEP 10 (2011) 038, arXiv:1105.4598 [hep-th].
  * (6) M. Cvetic, M. J. Duff, P. Hoxha, J. T. Liu, H. Lu, J. X. Lu, R. Martinez-Acosta, C. N. Pope, H. Sati, and T. A. Tran, “Embedding AdS black holes in ten-dimensions and eleven-dimensions,” Nucl. Phys. B558 (1999) 96–126, arXiv:hep-th/9903214 [hep-th].
  * (7) B. de Wit and H. Nicolai, “N=8 Supergravity,” Nucl.Phys. B208 (1982) 323.
  * (8) B. de Wit and H. Nicolai, “The Consistency of the S**7 Truncation in D=11 Supergravity,” Nucl. Phys. B281 (1987) 211–240.
  * (9) O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP 10 (2008) 091, arXiv:0806.1218 [hep-th].
  * (10) D. Z. Freedman and S. S. Pufu, “The holography of $F$-maximization,” JHEP 03 (2014) 135, arXiv:1302.7310 [hep-th].
  * (11) Y. Nakayama, “Topologically twisted renormalization group flow and its holographic dual,” Phys. Rev. D 95 no. 6, (2017) 066010, arXiv:1606.01988 [hep-th].
  * (12) Y. Nakayama, “Euclidean M-theory background dual to a three-dimensional scale-invariant field theory without conformal invariance,” Phys. Rev. D 95 no. 4, (2017) 046006, arXiv:1608.02651 [hep-th].
  * (13) S. B. Giddings and A. Strominger, “Axion Induced Topology Change in Quantum Gravity and String Theory,” Nucl. Phys. B 306 (1988) 890–907.
  * (14) N. Arkani-Hamed, J. Orgera, and J. Polchinski, “Euclidean wormholes in string theory,” JHEP 12 (2007) 018, arXiv:0705.2768 [hep-th].
  * (15) T. Hertog, M. Trigiante, and T. Van Riet, “Axion Wormholes in AdS Compactifications,” JHEP 06 (2017) 067, arXiv:1702.04622 [hep-th].
  * (16) D. Marolf and J. E. Santos, “AdS Euclidean wormholes,” Class. Quant. Grav. 38 no. 22, (2021) 224002, arXiv:2101.08875 [hep-th].
  * (17) G. J. Loges, G. Shiu, and T. Van Riet, “A 10d construction of Euclidean axion wormholes in flat and AdS space,” JHEP 06 (2023) 079, arXiv:2302.03688 [hep-th].
  * (18) D. Astesiano and F. F. Gautason, “Supersymmetric wormholes in String theory,” arXiv:2309.02481 [hep-th].
  * (19) N. Bobev, V. S. Min, K. Pilch, and F. Rosso, “Mass Deformations of the ABJM Theory: The Holographic Free Energy,” JHEP 03 (2019) 130, arXiv:1812.01026 [hep-th].
  * (20) P. G. Freund and M. A. Rubin, “Dynamics of Dimensional Reduction,” Phys.Lett. B97 (1980) 233–235.
  * (21) A. Azizi, H. Godazgar, M. Godazgar, and C. N. Pope, “Embedding of gauged STU supergravity in eleven dimensions,” Phys. Rev. D 94 no. 6, (2016) 066003, arXiv:1606.06954 [hep-th].
  * (22) A. Anabalón, M. Chamorro-Burgos, and A. Guarino, “Janus and Hades in M-theory,” JHEP 11 (2022) 150, arXiv:2207.09287 [hep-th].
  * (23) H. Elvang, R. Emparan, and A. Virmani, “Dynamics and stability of black rings,” JHEP 12 (2006) 074, arXiv:hep-th/0608076.
  * (24) M. Bianchi, D. Z. Freedman, and K. Skenderis, “Holographic renormalization,” Nucl. Phys. B 631 (2002) 159–194, arXiv:hep-th/0112119.
  * (25) The ambiguity of the sign in $N_{a}$ is because, in the wormhole solution, the boundary consists of two disconnected pieces located at $\mu=\pm\infty$.
  * (26) D. L. Jafferis, “The Exact Superconformal R-Symmetry Extremizes Z,” JHEP 05 (2012) 159, arXiv:1012.3210 [hep-th].
  * (27) Spinless fields in the $\textrm{SO}(8)$ supergravity transform in the $\bf{35}_{v}$ (proper-scalars) and $\bf{35}_{c}$ (pseudo-scalars) symmetric and traceless representations of $\textrm{SO}(8)$. The bosonic operator $\mathcal{O}_{S}$ in (46) corresponds to the trace of the boson bilinears and, therefore, has no associated scalar in the $\textrm{SO}(8)$ supergravity.
  * (28) See Table 1 of Freedman:2013oja  for a summary of the $\textrm{U}(1)_{\textrm{R}}\times\textrm{SU}(2)\times\textrm{SU}(2)\times\textrm{U}(1)_{b}$ charges of the various fields in the ABJM SCFT.
  * (29) A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,” JHEP 03 (2010) 089, arXiv:0909.4559 [hep-th].
  * (30) J. M. Maldacena and L. Maoz, “Wormholes in AdS,” JHEP 02 (2004) 053, arXiv:hep-th/0401024.
  * (31) E. Witten, “A Note On Complex Spacetime Metrics,” arXiv:2111.06514 [hep-th].
  * (32) A. Anabalón and J. Oliva, “Four-dimensional Traversable Wormholes and Bouncing Cosmologies in Vacuum,” JHEP 04 (2019) 106, arXiv:1811.03497 [hep-th].
  * (33) A. Anabalón, B. de Wit, and J. Oliva, “Supersymmetric traversable wormholes,” JHEP 09 (2020) 109, arXiv:2001.00606 [hep-th].
  * (34) N. Bobev, K. Pilch, and N. P. Warner, “Supersymmetric Janus Solutions in Four Dimensions,” JHEP 06 (2014) 058, arXiv:1311.4883 [hep-th].
  * (35) K. Pilch, A. Tyukov, and N. P. Warner, “$\mathcal{N}=2$ Supersymmetric Janus Solutions and Flows: From Gauged Supergravity to M Theory,” JHEP 05 (2016) 005, arXiv:1510.08090 [hep-th].
  * (36) G. Larios, P. Ntokos, and O. Varela, “Embedding the SU(3) sector of SO(8) supergravity in $D=11$,” Phys. Rev. D 100 no. 8, (2019) 086021, arXiv:1907.02087 [hep-th].