Classical and Quantum Gravity
Class. Quantum Grav. 38 (2021) 224002 (87pp) https://doi.org/10.1088/1361-6382/ac2cb7
AdS Euclidean wormholes
Donald Marolf1 and Jorge E Santos2,∗
1 Department of Physics, University of California at Santa Barbara, Santa Barbara,
CA 93106, United States of America
2 Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
E-mail: marolf@ucsb.edu and jss55@cam.ac.uk
Received 16 July 2021, revised 17 September 2021
Accepted for publication 4 October 2021
Published 26 October 2021
Abstract
We explore the construction and stability of asymptotically anti-de Sitter
Euclidean wormholes in a variety of models. In simple ad hoc low-energymod-
els, it is not hard to construct two-boundaryEuclideanwormholes that dominate
over disconnected solutions and which are stable (lacking negative modes) in
the usual sense of Euclidean quantumgravity. Indeed, the structure of such solu-
tions turns out to strongly resemble that of the Hawking–Page phase transition
for AdS–Schwarzschild black holes, in that for boundary sources above some
threshold we find both a ‘large’ and a ‘small’ branch of wormhole solutions
with the latter being stable and dominating over the disconnected solution for
large enough sources. We are also able to construct two-boundary Euclidean
wormholes in a variety of string compactifications that dominate over the dis-
connected solutions we find and that are stable with respect to field-theoretic
perturbations. However, as in classic examples investigated by Maldacena and
Maoz, the wormholes in these UV-complete settings always suffer from brane-
nucleation instabilities (even when sources that one might hope would stabilize
such instabilities are tuned to large values). This indicates the existence of addi-
tional disconnected solutions with lower action. We discuss the significance of
such results for the factorization problem of AdS/CFT.
Keywords: Page curve, Euclidean wormholes, string theory
(Some figures may appear in colour only in the online journal)
∗Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons
Attribution 4.0 licence. Any further distribution of this work must maintain attribution
to the author(s) and the title of the work, journal citation and DOI.
1361-6382/21/224002+87$33.00 © 2021 The Author(s). Published by IOP Publishing Ltd Printed in the UK 1
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
1. Introduction
It has long been understood that S-matrices, boundary correlators, or boundary partition func-
tions defined by bulk gravitational path integrals may fail to display familiar factorization
properties due to contributions from spacetime wormholes [1–6]. For our purposes, it is con-
venient to define spacetime wormholes as connected geometries whose boundaries have more
than one compact connected component. This definition includes real geometries of any sig-
nature as well as those that are intrinsically complex. Using an AdS/CFT language, the point is
that a boundary partition function Z is naively represented by a bulk path integral over config-
urations with a single compact boundary. Similarly, a product of boundary partition functions
(say, Z2) is naively represented by a bulk path integral over configurations with two discon-
nected boundaries. But contributions from spacetime wormholes suggest that the latter path
integral (which we call 〈Z2〉) is not necessarily the square of the former path integral (which
we call 〈Z〉); see figure 1.
Such failures of factorization would clearly require the standard picture (see e.g. [7–9]) of
AdS/CFT duality to be modified; see e.g. [4–6, 10, 11]) for related discussions. A possible
resolution is that the bulk path integral is in fact dual to an ensemble of boundary theories,
and our notation 〈Z〉, 〈Z2〉 is chosen to reflect this idea. A non-zero ‘connected correlator’
〈Z2〉 − 〈Z〉2 is then interpreted as describing δZ2 for fluctuations δZ that allow the partition
function Z in any particular element of the ensemble to differ from the ensemble-mean 〈Z〉.
Such an effective description was derived3 in [4–6] under certain locality assumptions, though
this assumption can be dropped by using the argument of [12]. In addition, dualities of this
kind have been explicitly constructed between (appropriate completions of) various versions
of Jackiw–Teitelboim (JT) gravity and corresponding double-scaled randommatrix ensembles
[13–15]. See also [16] for discussion of related issues for c = 1 matrix models, and [17] for
discussion of ensembles of Sachdev–Ye–Kitaev models [18, 19] and a proposed relation to
wormholes in JT gravity [20, 21] coupled to matter fields. Discussions of off-shell wormholes
in both JT gravity and pure gravity in AdS3 can be found in [22, 23]; see also [24, 25] for
related discussions of averaging 2D conformal field theories.
However, it is far from clear that this ensemble interpretation will hold for the most familiar
examples of AdS/CFT. In particular, such cases involve bulk theories with large amounts of
supersymmetry, and this supersymmetry should be reflected in each member of the boundary
ensemble4. However, in more than two boundary dimensions d the set of local boundary the-
ories with large amounts of supersymmetry is expected to be very limited; e.g., for d = 4
and N = 4 SUSY, super Yang–Mills theory is known to be the unique local maximally-
supersymmetric theory that admits a weakly-coupled limit5, and may thus be the unique
local theory. One might thus expect that—at least when the full physics associated with the
UV-completion of bulk gravity is taken into account—the ensemble of dual boundary the-
ories degenerates in this case so as to contain only a single physically-distinct theory (here
d = 4, N = 4 SYM); see related discussions in [10, 12, 27–29]. On the other hand, this would
require strong departures from the low-energy semi-classical description of the bulk and would
3While these works pre-date the discovery of AdS/CFT, their arguments apply immediately to that context.
4 This property holds in the matrix model examples of [15]. A general argument follows from the fact that the full bulk
system admits an algebra of asymptotic SUSY charges, and that these charges must act trivially on the ‘baby Universe
sector’ of the theory (i.e., on theHBU of [12, 26]). Thus the SUSY algebra acts within each bulk superselection sector.
The boundary dual interpretation is then that each member of the associated ensemble has a well-defined SUSY
algebra.
5 This follows from classifying the supersymmetric marginal deformations of free field theory.
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 1. An example showing failure of factorization due to spacetime wormholes.
The top line represents a path integral 〈Z〉. Although we have drawn the configuration
as connected, it may include contributions from disconnected spacetimes as well. In any
case, the natural path integral 〈Z2〉 associated with a pair of boundaries yields all terms
generated by squaring 〈Z〉, but also contains additional contributions connecting the two
boundaries as indicated by the second term in the bottom line.
thus render unclear the status of recent apparent successes [30–33] in using semi-classical bulk
physics to resolve issues in black hole information.
We thus return to the question of whether partition functions defined by higher-dimensional
bulk path integrals should factorize across disconnected boundaries; i.e., whether in such cases
we should in fact find 〈Z2〉 = 〈Z〉2. Such factorization would require either some rule or effect
to forbid spacetime wormholes from appearing in the path integral, or alternatively that in all
computations one finds precise cancellations when one sums over all connected spacetimes
with given disconnected boundaries. We note that, while it may at first appear somewhat con-
trived, the latter option is precisely what occurs in one of the baby-universe superselection
sectors described in [4–6, 12] and in the limit of the eigenbranes described in [34] where one
fixes all of the eigenvalues of the relevant matrices. Of course, in both of those cases one has
carefully tuned some extra ingredient (the baby Universe state or the eigenbrane source) to
make such cancellations occur. Furthermore, this option is difficult to see explicitly unless one
can solve the theory in detail.
It will come as no surprise that a complete study of the higher dimensional gravitational bulk
path integral is far beyond the scope of this work.We will therefore resort to the usual crutch of
studying bulk saddle points. Assuming that the contour of integration can be deformed to pass
through our saddles, they should dominate over contributions from non-saddle configurations
in the limit of small bulk Newton constant G.
Before proceeding, it is useful to review the literature concerning asymptotically AdS
spacetime-wormhole saddle-points. The simplest context in which one might imagine such
saddles to arise would be Euclidean pure AdS–Einstein–Maxwell theory with two spherical
boundaries (each Sd for a d + 1-dimensional bulk). However, such solutions are forbidden
by the results of [35], which showed that spacetime-wormhole saddle-points cannot arise in
Euclidean pure AdS–Einstein–Maxwell theory [35] when the scalar curvature of the boundary
metric is everywhere positive. On the other hand, although none of these solutions have all of
the properties that one might desire, a variety of Euclidean spacetime-wormhole saddles have
been constructed by allowing the boundary metric to be negative or by adding certain types of
matter [10, 27, 28] (with the latter based on the zero cosmological constant constructions of
[36]); see also [17] in the context of JT gravity with matter. We also refer the reader to the very
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
interesting story of constrained instantons (off-shell wormholes) described in [37], though at
least as of now their stability has been analyzed only in theories of pure gravity (and JT gravity
for d = 2).
In particular, the spacetime-wormhole solutions of [10, 27, 28] may be divided into four
categories. The first are the Euclidean solutions of [28] which have spherical (and thus positive-
curvature) boundaries but follow [36] in using axionic matter. The axion kinetic term can
become negative in Euclidean signature, though in this case it is positive definite but has a
surprising zero at a radius that depends on angular momentum. This unusual kinetic term and
a potential that is negative in the regionwhere the kinetic term is small combine to allow saddles
to suffer from bulk negative modes associated with non-trivial bulk angular momentum6 [38].
It is thus hard to argue that the dominate over non-saddles even at small G. Other spacetime
wormholes with field-theoretic bulk negative modes were described in section 4.2 of [10].
The second are Euclideanwormholeswith negative-curvatureboundaries (perhaps compact
hyperbolic manifolds). As discussed in [35], in the simplest AdS/CFT examples such solu-
tions have string-theoretic negative modes associated with the nucleation of D-branes. Indeed,
these negative modes render the entire theory unstable in the UV. While the UV issues can
be stabilized by appropriately deforming the CFT [10] (and, in particular, breaking conformal
invariance), and while this will forbid any brane nucleation instability close to the AdS bound-
ary, it was found in [27] that—at least in the model studied there—the wormholes remain
unstable to the nucleation of D-branes at finite locations in the interior.
The third class consists of Euclidean wormholes that have no known negative modes, and
which can even have lower action than disconnected solutions, but which have no known
embedding in string theory. Examples include the large α solutions in section 4.2 of [10].
Finally, the fourth category (section 5 of [10]) contains Euclidean wormholes with known
string-theoretic embeddings and no known negative modes, but where the corresponding
disconnected solution is not yet known so that it is unknown which saddle dominates at
large G.
We should also mention that the so-called ‘double-cone’ solutions of [13] define a 5th class
of spacetime wormhole solutions. These solutions are constructed by starting with e.g. the
complexification of a two-sided AdS–Schwarzschild black hole and taking the quotient by a
discrete Lorentz-signature time translation. The real Lorentz-signature section of this quotient
is connected and has two compact boundaries, though it is also singular at the bifurcation
surface. However, there are non-singular complex sections (on which the quotient acts freely)
that can be used to connect the same two real boundaries. As a result, we prefer to think of
the double-cone as an inherently complex solution. This is in no way a fundamental problem,
but we will instead discuss real Euclidean solutions below. We hope to return later to a more
detailed analysis of stability in the (complex) double cones.
Our goal here is thus to expand the class of known Euclidean spacetime wormholes with
simple disconnected boundary metrics, which we will take to consist either of two copies of
a (possible squashed) sphere (Sd) or two copies of the torus (Td). In particular, we wish to
identify saddles without field-theoretic negative modes, where the saddles can be embedded in
simple compactifications of string theory, and where the Euclidean wormholes dominate over
disconnected saddles. We will also explore negative modes associated with brane nucleation,
and we will analyze the structure of any the phase transition associated with the exchange of
dominance between Euclidean wormholes and disconnected saddles.
6 The situation is even worse if one makes a further analytic continuation of the axion according to φ→ iφ, as the
kinetic term then becomes negative definite. One might expect the same to be true in a reformulation writing the axion
in terms of the Hodge-dual two-form potential, but we have not carried out a detailed analysis.
4
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
We begin in section 2 by describing how a general class of potential Euclidean worm-
holes may be interpreted as homogeneous isotropic Euclidean cosmologies; i.e., as Euclidean
Friedmann–Lemaitre–Robertson–Walker (FLRW) solutions. This point of view provides use-
ful intuition for why Euclidean wormholes with positive curvature boundaries are forbidden
in pure AdS–Einstein–Hilbert gravity, and also for what sort of ingredients are required to
overcome this obstacle.
Sections 4 and 5 then study toy models in four bulk dimensions with simple bulk matter
content that concretely illustrate the construction suggested by the FLRW analysis of section 2
with S3 boundaries. As examples of bulk matter we consider bothU(1) gauge fields (section 4)
and non-axionic scalar fields (section 5). These models do not directly embed in string theory,
but turn out to be similar to some that do. In the toy models we identify connected (wormhole)
solutions that are free of bulk field-theoretic negative modes and which dominate over discon-
nected (non-wormhole) solutions in appropriate regimes. As a function of the matter sources,
we find the associated space of solutions to have structure much like that of the well-known
Hawking–Page transition, including in particular a first-order phase transition associated with
exchanging dominance between the connected and disconnected saddles. This is precisely the
structure that was found in studies of wormholes in JT gravity coupled to matter [17] and in
studies of constrained wormholes [37] in theories of pure gravity. Because these are only ad
hoc low-energy theories or JT models, they do not contain fundamental branes. Thus there can
be no notion of brane-nucleation instability to explore in such models.
We therefore turn in sections 6–8 to studying truncations of string-theory or M-theory. In
particular, section 6 considers a truncation of 11-dimensional supergravity that reduces toU(1)2
Maxwell–Einstein theory on AdS4, section 7 examines a mass-deformed version of the ABJM
theory [39], and section 8 investigates a truncation of type IIB string theory to an Einstein-
scalar system on AdS4. In each of these cases we require spherical boundaries and construct
wormhole solutions. The structure of the space of solutions is generally much as in the toy
models, with wormholes dominating over the most obvious disconnected solution at large val-
ues of the relevant boundary source, and with such wormholes being free of field-theoretic
negative modes. However, in all cases we find such would-be dominant wormholes to suffer
from brane-nucleation instabilities. In sections 6 and 8 the brane-nucleation occurs only in a
finite region of the bulk and does not occur in the deep UV. Thus the theory itself remains stable
with these boundary conditions and only the particular wormhole solution is destabilized. In
contrast, section 7 studies a context with hyperbolic boundaries of the form reviewed above,
but with a deformation parameter μ that is expected to stabilize the theory at large enough μ.
While it does in fact appear to do so, we find wormholes only in the small-μ regime where
the theory remains unstable. The results of section 7 are thus similar to those found in [27] for
mass deformations ofN = 4 SYM.
We close with some interpretation and discussion of open issues in section 9. The main
text is supplemented by various appendices with additional technical details. This includes
appendix C, which describes studies of two UV-complete models with torus boundaries (with
results broadly similar to those associated with spherical boundaries). It also includes appendix
D, which lists results for a larger set of 14 low energymodels and 22 string/M-theory compacti-
fications in a variety of dimensions that we have studied at least briefly but whose investigation
we may not have chosen to describe in detail. While the explorations of those models were not
always as complete as the ones described in the main text (see appendix D for details), they
suggest that the results of sections 6–8 are typical.
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
2. FLRW approach
Consider any Euclidean wormhole whose boundary consists of two copies of a maximally-
symmetric Euclidean geometry Σd; i.e., where Σd is a sphere S
d, a Euclidean plane Rd, or
a hyperbolic plane Hd. If the metric of the full bulk solution preserves this symmetry, then
the wormhole admits a preferred slicing which again preserves the symmetry. We may thus
describe the wormhole as a d + 1-dimensional homogeneous isotropic Euclidean cosmology
with Sd , Rd , or Hd slices and with Euclidean time running transverse to each slice.
We may thus write the metric in the (Euclidean) FLRW form
ds2 = dτ 2 + a2(τ )dΣ2d, (2.1)
with dΣ2d the standard metric on S
d,Rd , orHd, where in the case of Sd orHd we take the spaces
to have unit radius for simplicity. Labeling the three cases by k = 1, 0,−1 as usual, it is well
known that the Einstein equation
Rab − gab2 R = 8πGTab (2.2)
with Tab the stress energy tensor for bulk matter, reduces to the Friedman equation7(
a˙
a
)2
= − 16πG
d(d − 1)ρ+
1
L2
+
k
a2
. (2.3)
Here a˙ = dadτ , L is the bulk AdS scale and the term
1
L2 encodes the explicit effects of the negative
cosmological constant, and ρ is the standard (Lorentz-signature) energy density of any matter
fields
ρ ≡ −Tττ . (2.4)
Due to the Euclidean signature this equation differs from its familiar Lorentz-signature coun-
terpart by an overall sign. Of course, we can also take quotients of the above solutions and thus
use this formalism when Σd is a torus Td or a compact hyperbolic manifold Hd/Γ for some
properly discontinuous isometry group Γ of Hd.
Let us now briefly investigate what (2.3) implies for the existence of Euclidean wormholes.
For τ →±∞, we ask that a(τ )→∞ to satisfy asymptotically AdS boundary conditions. As a
result, on any wormhole solution a(τ ) must have some minimum a0 where a˙ = 0. This clearly
requires the right-hand side of (2.3) to vanish. Without matter we have ρ = 0, so this condi-
tion can be satisfied only when k = −1; i.e., when the boundary metric has negative scalar
curvature.
However,we immediately notice two encouraging features. First, for k = 0 the above failure
is only marginal. Multiplying (2.3) by a2 and setting k = 0, we see that a˙ can vanish at a =
0. Indeed, there is a k = 0 vacuum solution a(τ ) = L eτ/L. One may think of two copies of
this solution as describing a degenerate limit of Euclidean wormholes where the neck of the
wormhole has been stretched to become both infinitely long and infinitely thin.
Second, for any k it is clear that the a˙ = 0 constraint can be satisfied by adding matter with
positive Lorentz-signature energy ρ. For k = 0 an arbitrarily small amount of such matter will
do, but for k = 1 we require ρ to exceed a critical threshold. We thus expect k = 0 wormholes
7 The so-called second Friedman equation follows from the time derivative of (2.3) and conservation of stress-energy.
6
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
to appear with arbitrarily small matter sources on the boundaries, while for k = 1 wormholes
will arise only when the scalar sources are sufficiently large.
Many familiar kinds of matter yield positive Lorentz-signature energy densities ρ. However,
especially since the matter energy density ρwill be large for k = 1, it is useful to choosematter
which is gravitationally attractive in Lorentz signature. Wick rotation to Euclidean signature
then gives gravitational repulsion, which helps to make a¨ positive at a = a0 and also helps to
satisfy the asymptotically AdS boundary conditions. In particular, it would not be useful to
use the potential energy of a scalar field, where the condition ρ > 0 would effectively require
adding a new positive cosmological constant to cancel the old (negative) one.
Furthermore, if the energy at a = a0 comes from time-derivatives, then it is naturally posi-
tive if the Lorentz-signature field is real. But that will make τ -derivatives imaginary at a0, and
thus tend to give imaginary (or complex) fields in Euclidean signature. The kinetic terms of
such fields then tend to give negative contributions to the Euclidean action, and are thus a likely
source of negative modes. This is the essential reason why the axion solutions of [28, 36] have
many negative modes8 [38]. We thus wish to take all τ -derivatives to vanish at a0. Assuming
that the scalar sources are identical on the two boundaries, this is equivalent to requiring the
entire wormhole to be invariant under a correspondingZ2 symmetry.
As a result, we study solutions below with a surface a = a0 invariant under such a Z2
symmetry and where ρ > 0 at this surface due to spatial gradients of the matter fields. Such
kinetic energy is indeed gravitationally attractive in Lorentz signature, and thus gravitationally
repulsive in Euclidean signature, but is consistent with real Euclidean solutions. Creating such
gradients requires similar gradients in the scalar sources we choose at the two boundaries. For
k = 0, we expect Euclidean wormhole solutions with arbitrarily small such sources. For k = 1,
we expect Euclidean wormhole solutions to appear once the boundary sources exceed some
critical threshold.
Note that the above analysis and statement of expectations applies only when the surfaces
of constant Euclidean time are homogeneous and isotropic. Since the matter fields have spatial
gradients, we will need to choose a finely-tuned matter ansatz to achieve this. We may expect
similar behavior for more general solutions that violate homogeneity9, but finding solutions
would then require the solution of partial differential equations. We thus save such analyses
for future work.
3. The general strategy for analysing field theoretical negative modes
Let us consider a general Euclidean partition function Z associated with a Euclidean action
SE
[
φ; φ∂M
]
with some collection of fields φ and corresponding boundary conditions φ∂M. For
the case of an Einstein-scalar theory, φwould contain all (d + 1)(d + 2)/2 independent metric
components and the scalar field. Such a partition function can be schematically represented as
Z[φ∂M] =
∫
Dφ e−SE
[
φ;φ∂M
]
. (3.1)
8 As discussed above, the kinetic term of [38] does not become negative. But it does have a surprising zero.
9 For k = 0, spacetime wormholes should require non-zero gradients along each leg of the torus. Otherwise one could
Wick-rotate along a leg with translational symmetry and find a Lorentz-signature solution with two disconnected non-
interacting boundaries linked by a traversable wormhole (and thus violating boundary causality). We thanks Douglas
Stanford for discussions on this point.
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
In the saddle-point approximation, we can expand Z as
Z[φ∂M] ≈ e−SE
[
φ 0;φ∂M
]
×
∫
D δφ e−S
(2)
E
[
δφ;0
]
+ · · · , (3.2)
where φ0 are classical solutions of the equations of motion derived from SE
[
φ; φ∂M
]
, and we
obtain S(2)E
[
δφ; 0
]
by expanding the fields as φ = φ0 + δφ and keeping all terms up to second
order in δφ. A first order term is absent in the expansion above, because φ0 satisfies the classical
equations of motion derived from SE
[
φ; φ∂M
]
.
Let us imagine for a moment that S(2)E
[
δφ; 0
]
is not positive definite. In that case the saddle
φ0 is not a local minimum of the Euclidean action and does not dominate over integration over
nearby configurations if the integral is performed along the real Euclidean contour. In this case
we say that φ0 has field-theoretic negative modes.
For pure gravity S(2)E
[
δφ; 0
]
is infamously not positive definite [40, 41]. In fact one can
show that the conformal factor of the metric always has the wrong sign for the kinetic term.
This is the conformal factor problem of Euclidean quantum gravity. One way around this is to
Wick rotate the conformal factor, leading to a convergent Gaussian integral. This procedure,
although slightly ad hoc in [40, 41], was justified at the level of linearized gravity in [42] and
has been recently backed up by detailed dual field theory calculations [43–45] in the context
of gauge/gravity duality. It was also shown in [46] that a version of this Wick rotation can be
performed at the non-linear level.
The case of gravity coupled to matter is more delicate. In particular, it is no longer obvious
that the conformal factor is the right variable to Wick rotate [47] since perturbations of the
conformal factor, i.e. trace-type metric perturbations, will generically couple to other scalar
matter perturbations. In addition, if matter is present, the trace free part of the metric can also
couple with the trace itself.
Here we follow the procedure outlined in [48] which was used in [47] to investigate the
negative mode of an asymptotically flat Reissner–Nodström black hole. It turns out that in all
the cases we studied, the action can be decomposed as
S(2)E
[
δφ; 0
]
= Sˆ(2)E
[

δφˆ; 0
]
+ S˜(2)E
[
δφ˜; 0
]
, (3.3)
where together the perturbations δφˆ and δφ˜ span the space of the original perturbations δφ
and both Sˆ(2)E
[

δφˆ; 0
]
and S˜(2)E
[
δφ˜; 0
]
have been written in first order form. The variables δφˆ
turn out to be non-dynamical, i.e. the action Sˆ(2)E
[

δφˆ; 0
]
contains no derivatives of δφˆ. It is
in this sector that we find a mode with a non-positive action. Furthermore, the fact that the
variables δφˆ enter the action algebraically is a consequence of the Bianchi identities, and in
an appropriate canonical formalism these variables would become Lagrange multipliers that
enforce constraints. Let us denote by { δφˆ}0 the problematic mode. This is the mode that we
Wick rotate as { δφˆ}0 → i { δφˆ}0. The Gaussian integral over δφˆ can now be performed and we
reabsorb it in the measure. We are then left to study the positivity properties of S˜(2)E
[

δφ˜; 0
]
. At
this stagewe introduce gauge invariant variables ˇqwhich can be used to write S˜(2)E
[

δφ˜; 0
]
solely
as a function of ˇq and their first derivatives. It is then S˜(2)E
[
ˇq; 0
]
whose positivity properties we
8
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
investigate. Note that the dimensionality of ˇq is necessarily smaller than that of δφ˜ because of
gauge invariance.
The procedure outlined above is consistent with the studies performed in [49–51] which
were used in [38] to show the existence of multiple negative modes on wormholes sourced by
axions; see also [52].
4. Einstein-U(1)3 theory with S3 boundary
We now proceed to study a simple AdS4 Einstein–Maxwell model that illustrates both key
elements of the physics and our main techniques. We assume spherical symmetry, and in par-
ticular a spherical boundary metric. We begin with an overview of our model and then discuss
disconnected solutions in section 4.1, and connected wormhole solutions in section 4.2. In
particular, we will see that connected wormholes can have lower action than the disconnect
solution. Finally, we show in section 4.3 that these low-actionwormholes are stable in the sense
that they have no Euclidean negative modes. Since this is merely an ad hoc low-energy model
not derived from string theory (or any other UV-complete theory), the discussion in section 4.3
concerns only field-theoretic negativemodes. There is no possible notion of a brane-nucleation
negativemodes as the theory does not contain branes (nor does it contain non-singularmagnetic
monopoles).
As described below, choosing themodel to contain three distinctMaxwell fields will help us
to arrange a cohomogeneity-1 solution; i.e., a solution that is homogeneous at each ‘Euclidean
time’ in the FLRW sense described above.We will thus search for wormhole and disconnected
solutions to the equations of motion derived from the following action:
SU(1)3 = −
∫
M
d4x
√
g
(
R+
6
L2
−
3∑
I=1
F(I)abF
(I) ab
)
− 2
∫
∂M
d3x
√
h K + SB,
(4.1)
where L is the AdS length scale, K is the trace of the extrinsic curvature associated with an
outward-pointing normal to ∂M, h the determinant of the induced metric hμν on ∂M and
F(I) = dA(I). Here and throughout this paper will take units in which 16πG = 1.
The second term in (4.1) is the so-called Gibbons–Hawking–York term. The final term
SB includes a number of counterterms that render the Euclidean on-shell action finite and are
functions of the intrinsic geometry on ∂M only and are dimension dependent. For the above
theory in four bulk spacetime dimensions these turn out to be given by
SB =
4
L
∫
∂M
d3x
√
h+ L
∫
∂M
d3x
√
hR, (4.2)
where R is the intrinsic Ricci scalar on ∂M. One might wonder whether we need additional
boundary terms associatedwithF(I) such as the ones reported in [53]. However, as noted in [53],
no such terms are needed if we are interested in fixing the leading value of A(I) as we approach
the conformal boundary, i.e. work in the grand-canonical ensemble. These are precisely the
boundary conditions we choose. The equations of motion derived from (4.1) read
Rab +
3
L2
gab = 2
3∑
I=1
(
F(I)ac F
(I)c
b −
gab
4
F(I)cd F
(I)cd
)
, (4.3a)
9
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
∇aF(I)ab = 0. (4.3b)
subject to the boundary conditions that on ∂M, the induced metric h is fixed as well as A(I).
We are interested in finding solutions for which the metric has the same isometries as a
round three-sphere, i.e. spherical symmetry, and in particular SO(4), but where the Maxwell
fields explicitly break such symmetry.An easy way to do so is to write the three-sphere in terms
of left-invariant one-forms{σˆ1, σˆ2, σˆ3} such that themetric on the unit round three sphere reads
dΩ23 =
1
4
(
σˆ21 + σˆ
2
2 + σˆ
2
3
)
, (4.4)
with
dσˆI =
1
2
εIJK σˆJ ∧ σˆK . (4.5)
In terms of standard Euler angles, we can choose
σˆ1 = − sin ψ dθ + cos ψ sin θ dϕˆ (4.6a)
σˆ2 = cos ψ dθ + sin ψ sin θ dϕˆ (4.6b)
σˆ3 = dψ + cos θ dϕˆ (4.6c)
with ψ ∈ (0, 4π), θ ∈ (0, π) and ϕˆ ∈ (0, 2π).
We then search for solutions of the form
ds2 =
dr2
f (r)
+ gS3 (r)dΩ
2
3. (4.7)
For the Maxwell fields, we choose
A(I) = L
σˆI
2
Φ(r), for I ∈ {1, 2, 3}. (4.8)
4.1. Disconnected solutions
A primary question will be whether connected wormhole solutions dominate over discon-
nected solutions. We thus begin here by constructing the simpler disconnected solution for
comparison, deferring discussion of wormhole solutions to section 4.2 below.
As is often the case, it is convenient to fix the gauge in equation (4.7) by choosing
gS3 (r) = r
2. (4.9)
We take r ∈ (0,+∞), with r = 0 describing the smooth center where the round S3 shrinks
smoothly to zero size and r = +∞ the location of the asymptotic conformal boundary.
Regularity of F(I) at the origin demands that
dΦ(r)
dr
∣∣∣∣
r=0
= 0, (4.10)
with Φ(0) being a constant, whereas regularity of the metric at the point r = 0 demands
f (0) = 1. (4.11)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Note that regularity of A(I) seen as a one-form demands that Φ(r) = O(r2), which is stronger
than the condition expressed by equation (4.10).
With our choice of boundary conditions, we find a unique solution given by
f (r) = 1+
r2
L2
and Φ(r) = Φ0
√
L2 + r2 − L√
L2 + r2 + L
. (4.12)
It turns out that eachmember of this one-parameter family of solutions is self-dual, in the sense
that
F(I) = F(I), (4.13)
where  is the Hodge dual operation in four spacetime dimensions. Recall that for self-dual
solutions the stress energy tensor induced by F(I) is identically zero, which is why the metric
is identically Euclidean AdS for any value of Φ0. Note also that the above solution hasΦ(r) =
O(r2) near r = 0 so that our boundary condition is satisfied.
It is straightforward to evaluate the Euclidean on-shell action on these solutions. It must of
course be a function of Φ0 only, and we find the particular form
SDU(1)3 = 8π
2L2
(
1+ 3Φ20
)
. (4.14)
Here the upper-script D on the left-hand-side denotes the on-shell action of the disconnected
solution.
4.2. Wormhole solutions
Having found our disconnected solution, we now turn to the study of smooth connected worm-
holes. Since such solutions will have a minimal sphere of some non-zero area 4πr20, we now
choose to fix the gauge in equation (4.7) by writing
gS3 (r) = r
2 + r20. (4.15)
Now r ∈ R, with the two asymptotic boundaries located at r→±∞. We also impose a global
Z2 symmetry that relates the two spheres of given r 
= r0 and which leaves the minimal sphere
fixed.We shall see that the parameter r0 will be a function ofΦ0 only.Without loss of generality
we will take r0 > 0.
Again, we can integrate our equations of motion to find the full space of such solutions.
Note that our Z2 symmetry requires
dΦ(r)
dr
∣∣∣∣
r=0
= 0. (4.16)
From the equations for the Maxwell fields we find
f (r) =
C + 4Φ(r)2
(r2 + r20)Φ
′(r)2
, (4.17)
where C is a constant to be determined later and ′ denotes differentiation with respect to r.
Consistency of the rr and S3S3 components of the Einstein equation then demands
Φ′(r)2 − L
2r2
[
C + 4Φ(r)2
](
r2 + r20
) [
CL4 +
(
r2 + r20
) (
L2 + r2 + r20
)] = 0. (4.18)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Since this gives an explicit result for [C + 4Φ(r)2]/Φ′ it allows us to write f (r) in the form
f (r) =
CL4 +
(
r2 + r20
) (
L2 + r2 + r20
)
L2r2
. (4.19)
Thus, we find a singularity at r = 0, unless we set
C = −L
2r20 + r
4
0
L4
, (4.20)
With this choice for C one obtains
f (r) =
L2 + r2 + 2r20
L2
, (4.21)
which is smooth at r = 0 as desired. Using our choice of the constant C in (4.18) yields
Φ′(r)2 − 4L
4Φ(r)2 − r20
(
L2 + r20
)
L2
(
r2 + r20
) (
L2 + r2 + 2r20
) = 0. (4.22)
Since we require Φ′(0) = 0 at r = 0 we want to impose, the above fixes r0 in terms of
Φ(0) ≡ Φ to be
r0 = b L with a ≡ (1+ 16Φ2)1/4 and b ≡
√
a2 − 1
2
. (4.23)
With these choices, the equation for Φ can be readily solved to give
Φ(r) = Φ cosh
[
2
b
F
(
arctan
( r
La
)∣∣∣ 1− a2
b2
)]
, (4.24)
where F(φ|m) is the elliptic integral of the first kind.
To determine the source Φ0 in terms of Φ we simply expand the above Φ(r) at large r, to
find
Φ0(Φ) = Φ cosh
[
2
b
K
(
1− a
2
b2
)]
, (4.25)
where K(m) is the complete elliptic integral of the first kind. We stress that Φ0 is the actual
chemical potential for A(I), but that we find it more convenient to parameterize the solutions in
terms ofΦ. The reason for this is that there can be more than one solution for a given value of
Φ0, but that solutions are uniquely determined by their value of Φ∗. This is best illustrated by
looking a plot of Φ0(Φ) (see figure 2). From this plot it is clear that wormhole solutions can
only exist if Φ0  Φmin0 ≈ 3.563349, for which Φ = Φmin ≈ 1.002373.
But what distinguishes the two wormholes with a given value of Φ0? Perhaps the best way
to see the answer is to study the radius r0 of the wormhole throat as a function of Φ0 shown in
figure 3. For any fixed value of Φ0 > Φ
min
0 ≈ 3.563349, two wormhole solutions exist: a large
wormhole and a small wormhole. For Φ0 = Φ
min
0 we have r0 = r
min
0 ≈ 1.251462L, and both
the large and small branches merge. We shall see that these solutions behave just like small
12
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 2. The source for the Maxwell fields A(I)Φ0 as a function of Φ. There is a min-
imum value of Φ0, Φmin0 ≈ 3.563349, above which two types of wormhole solutions
exist.
Figure 3. Radius of the wormhole solutions as a function of the Maxwell source Φ0.
For fixed value of Φ0 > Φ
min
0 two wormhole solutions exist.
and large Euclidean Schwarzschild black holes in global AdS. In particular, we will show that
the small wormhole branch has a negative mode, and the large wormhole branch does not.
13
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
One can also evaluate the Euclidean on-shell associatedwith our wormhole solutions, which
can be written in terms of complete elliptic integrals of the first and second kind in the form
SCU(1)3 =
8L2π2
(X − 1)3/2
[
2(X − 1)E(−X)− (X − 2)K(−X)
+
3X
4
√
X − 1 sinh
(
4
√
X − 1K(−X)
)]
. (4.26)
Here we defined
X ≡ 1+ L
2
r20
, (4.27)
and E(m) is the complete elliptic integral of the second kind.
We can finally plot one of our figures of merit, namely
ΔSU(1)3 ≡ 2SDU(1)3 − SCU(1)3 . (4.28)
IfΔSU(1)3 is positive, the wormhole solution has lower Euclidean action than the disconnected
solution with the same value of Φ0. If, on the other hand, ΔSU(1)3 < 0, it must be that the
wormhole solution is subdominant. We find that the large wormhole solutions are dominant
forΦ0 > ΦHP ≈ 3.859673, and subdominant otherwise.We denote the transition value byΦHP
due to the similarity to the familiar Hawking–Page transition. The small wormhole solutions
are always subdominant. These two behaviors are displayed in figure 4. A similar structure
was found previously for wormholes in JT gravity coupled to matter [17].
Having found dominant saddles, we now proceed to determine their stability.
We now note that all the solutions we found, either connected or disconnected in the bulk,
satisfy a Euclidean version of the first law involving the Euclidean action SE. According to
standard lore in AdS/CFT, one can find the expectation value of the operators dual to AI by
simply taking a functional derivative of the action with respect to the corresponding boundary
value of AI
〈J Iμ〉 =
δSE
δAIμ
, (4.29)
where Greek indices run over boundary coordinates. These can be easily evaluated on arbitrary
on shell solution and it turns out that
〈J Iμ〉 = 8π2L3J˜Iμ (4.30)
with J˜Iμ being given by
AI = AIa dx
a = ΦIμ dx
μ − zJ˜Iμ dxμ +O(z2), (4.31)
where z is a Fefferman–Graham coordinate [54]. This in turn implies that
dSE = εJIμ dΦ
Iμ, (4.32)
where ε = 1 for disconnected solutions and ε = 2 for wormholes with two boundaries. We
have checked that our solutions satisfy this relation. Perhaps more importantly, appropriate
generalisation of this type of first law also arise when studying scalar wormhole solutions,
whichwewere only able to study numerically.We have checked that all our numerical solutions
satisfy the above relations to better than 10−10% accuracy.
14
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 4. The difference in Euclidean action between the disconnected and connected
solutions. At any Φ0, the larger value ofΔSU(1)3 corresponds to the large wormhole and
the smaller value corresponds to the small wormhole. Due to the similarity to the familiar
Hawking–Page transition, we use ΦHP to denote the value of Φ0 at which ΔSU(1)3 = 0
for the large wormhole. For Φ0 > ΦHP, the large wormhole solution becomes dominant
while the small wormhole solution is always subdominant.
4.3. Negative modes
We now discuss perturbations around our wormholes, and in particular, the possible existence
of negative modes. We will take advantage of the SO(4) symmetry of the S3 to decompose
the perturbations into spherical harmonics. Perturbations then will come into three different
classes: scalar-derived perturbations, vector-derived perturbations and tensor-derived pertur-
bations. These are built from scalar, vector and tensor harmonics on the S3. We shall label
each of these structure functions by SS , SVi and S
T
i j , with S = 0, 1, 2, . . . , V = 1, 2, . . . and
T = 2, 3, . . . and i, j running over the sphere directions.
These structure functions are chosen so that they are orthogonal to each other in the absence
of background fields that might break the SO(4) symmetry. Unfortunately, the Maxwell fields
do break SO(4), so we will need more structure. Nevertheless, we will be able to use these
building blocks to study the negative modes. When there is SO(4) background symmetry of
the background, orthogonality only occurs if we take SVi to be divergence free and S
T
i j to be
traceless-transverse. All these operations are, of course, done with respect to the metric on the
round three-sphere. The scalars in addition satisfy
S3SS + λSSS = 0, (4.33a)
with λS = S(S + 2), the vectors
S3S
V
i + λVS
V
i = 0, (4.33b)
15
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
with λV = V(V + 2)− 1 and the tensors
S3STi j + λTS
T
i j = 0, (4.33c)
with λT = T (T + 2)− 2.
4.3.1. Scalar-derived perturbations: the S = 0 sector. This is the only sector where we find
a negative mode, and it occurs only for the small wormhole branch. In fact, the threshold
for the existence of this mode coincides precisely with r0 = rmin0 . This is akin of what hap-
pens with Schwarzschild–AdS, where a negative mode exists for small black holes, but not
for large black holes [55]. The threshold can be found analytically by inspecting when the
Schwarzschild–AdS black holes become locally thermodynamically stable, i.e. when the spe-
cific heat becomes positive. Note that in Schwarzschild–AdS this transition occurs before the
Hawking–Page transition, so that when the large black hole branch dominates over pure ther-
mal AdS, the negative mode is no longer presence. We shall see a similar behavior with the
spherical wormholes.
We start with an ansatz for the  = 0 sector. Since there are no vector or tensor perturbations
on the S3 with  = 0, our ansatz preserves the same symmetries as the background solution.
We thus search for negative modes which take the same form as equations (4.7) and (4.8) with
g(r) = g+ δg(r), f (r) = f + δ f (r) and Φ(r) = Φ(r)+ δΦ(r), (4.34)
where Φ is given in equation (4.24) and
g = r2 + r20, and f =
L2 + r2 + 2r20
L2
. (4.35)
We expand the action (4.1) to second order in δg, δ f and δΦ. The terms linear in δg, δ f and δΦ
vanish by virtue of the background equations ofmotion. It is then a simple exercise to recast the
action as a function of δg, δ f and δΦ and their first derivatives only. Doing so involves integrat-
ing by parts, and the resulting boundary term precisely cancels the Gibbons–Hawking–York
term. We shall denote this quadratic action by S(2). Furthermore, we find that δ f enters the
action algebraically as it should by virtue of the Bianchi identities. This makes it straight-
forward to integrate out δ f (since the action is quadratic in δ f and thus the path integral is
Gaussian). The result of this procedure yields an action that is quadratic in δΦ and δg and their
first derivatives. We have also checked that the path integral in δ f has the correct sign for a
meaningful integration, i.e. the coefficient of the term proportional to δ f 2 is negative definite.
We denote the resulting action by S˜(2).
We wish to work with gauge invariant perturbations. In order to do this, we must first under-
stand how δg, δ f and δΦ transformunder an infinitesimal gauge transformation ξ = ξr(r)∂/∂r.
This is easily seen by recalling that a metric perturbation h and gauge field perturbation a,
transforms under infinitesimal gauge transformations ξ as
Δh = £ξ g¯, Δa = £ξA¯, (4.36)
where £ξ is the Lie derivative along ξ, and (g,A) are the metric and gauge potential background
fields.
We then find
Δδ f = ξr f¯
′ − 2 f¯ ξ′r, Δδg = 2rξr and ΔδΦ = ξrΦ¯′. (4.37)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
As a result, we can then build the gauge invariant quantity
q = δΦ− Φ′ δg
2r
. (4.38)
It is a trivial exercise to show that Δq = 0. We can use this definition to write the quadratic
action S˜(2) for δg and δΦ as a function of q. This is done by effectively solving equation (4.38)
for δΦ and substituting the resulting expression for δΦ in the quadratic action S˜(2). The depen-
dence in δg completely cancels out, as it should, due to gauge invariance.We are thus left with
S˜(2) written in terms of q, q and its first derivative only:
S˜(2) = 2π2
∫ ∞
−∞
dr
√
g
f
[
f Kq′2 + Vq2
]
, (4.39a)
where
K =
6L2r2
r2 − gL2Φ′2
and V =
4K
g
[
2g
L2 f r
L2
(
r − L2ΦΦ′)+ g (r − 2L2ΦΦ′)
r2 − gL2Φ′2
− 1
]
.
(4.39b)
Two comments regarding boundary terms are now in order. First, to show that δg drops out
one needs to integrate by parts twice. This generates two boundary terms. These two terms
precisely cancel the counterterms in equation (4.1). Second, in order to ensure that terms pro-
portional to qq′ do not show up in the final form of the action, we again had to integrate by
parts. It is easy to show that the resulting boundary terms vanish so long as q ∼ o(r4) near the
conformal boundary. As we shall see below, our boundary conditions for q will require that q
vanishes at this rate near the boundary, so these terms can be safely neglected.
To search for negativemodes, we integrate the first term in equation (4.39a) by parts to write
S˜(2) = 2π2
∫ ∞
−∞
dr
√
g
f
q
⎧⎨
⎩−
√
f
g
[√
f gKq′
]′
+ Vq
⎫⎬
⎭ . (4.40)
The resulting boundary term can be neglected so long as q ∼ o(r−1/2), and wewill verify below
that such boundary conditions may be imposed. The negative mode equation simply becomes
−
√
f
g
[√
f gKq′
]′
+ Vq = λq. (4.41)
If we can find values of λ < 0 for which this equation admits non-trivial solutions, then fluc-
tuations about the saddle will make large contributions and the Euclidean solution is locally
unstable. Finally, we still need to checkwhether the possible behaviors that this equation admits
near the conformal boundary are consistent with imposing a boundary condition that requires
q ∼ o(r−1/2), so that we can indeed neglect the above boundary terms. A Frobenius analysis
[56] close to the conformal boundary reveals that
q ∼ r−Δ± with Δ± = 12 ±
√
1
4
− λ
6
. (4.42)
We see that for λ < 0 theΔ+ branch satisfies q ∼ o(r−1/2). It is thus consistent to require this
as a boundary condition and to then neglect the above-mentioned boundary terms.
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 5. The homogeneous negative mode with  = 0 as a function of the black hole
radius, measured in units of rmin0 . At r = r
min
0 the negative mode vanishes, and becomes
positive thereafter.
Since V(r) is symmetric around r = 0, we can decompose our modes into modes with
q(0) = 0 or q′(0) = 0. For numerical convenience we also define
q =
LΔ+
(r2 + r20)
Δ+/2
qˆ and r =
r0y
1− y (4.43)
so that the conformal boundary is located at y = 1 and the origin at y = 0. Solving
equation (4.42) off the conformal boundary yields qˆ′(1) = 0 for the choice q ∼ r−Δ+ .
Using the numerical methods first outlined in [57] and reviewed in [58] we search for neg-
ative modes with the above boundary conditions. For qˆ(0) = 0 we find no negative modes for
any value of r0/L. On the other hand, for qˆ′(0) = 0 we find exactly one negative mode, which
becomes positive when r = rmin0 (see figure 5). This is precisely when the transition between
small and large wormholes occurs, in complete analogy with spherical Schwarzschild–AdS
black holes.
4.3.2. Scalar-derived perturbations: the S  2 sector. Here we partially follow the pioneer-
ing work of Kodama and Ishibashi in [59]. For the metric we take our perturbations to have the
form
δ ds2S = h
S
rr (r)S
S dr2 + 2hSr (r)∇ i SS dr dyi
+ HST (r)S
S
i j dy
i dy j + HSL (r)S
S𝕘i j dyi dy j, (4.44)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
where∇ is the covariant derivative on S3, y are coordinates on the S3, lower case Latin indices
live on S3, 𝕘 is the round metric on S3 and
S
S
i j = ∇ i ∇ j SS −
𝕘i j
3
∇ k ∇ kSS . (4.45)
Here SSi j is by construction trace free. The perturbation of the gauge fields is more intricate.
Since the background Maxwell fields break the full SO(4), we expect that their perturbations
will strongly depend on the background field. Note that SSi j vanishes identically for S = 1 so
that this mode must be treated separately; see section 4.3.3.
It is our objective to list all independent vector fields on S3 that are linear in SS and can be
built with the background fields AI and metric 𝕘. These turn out to be
δASI = A
S(r)SSAI + B
S(r)£˜ΞS AI + C
S(r)AiI ∇ i SS dr
+ DSI (r)dr + E
S
I (r)∇ i SS dyi, (4.46)
where £˜ is the Lie derivative acting on S3 and
ΞS = ∇ i SS ∂
∂yi
. (4.47)
One might think that one would need to include additional terms of the form
ιΞS dAI , (4.48)
but these can be reabsorbed into the coefficients AS ,BS ,CS with a U(1) gauge transformation
on δASI .
In order to check that our ansatz is nontrivial, we have linearized the corresponding Ein-
stein–Maxwell equations and found that there are three dynamical second order gauge invariant
equations that govern such perturbations. Since we want to work with gauge invariant pertur-
bations, we must find how these perturbations transform under infinitesimal gauge transforma-
tions (both U(1) and diffeomorphisms). In order to do this, we need to sort out how to write
infinitesimal diffeomorphisms in terms of the scalar harmonics SS . Let ξS be the infinitesimal
diffeomorphism associated with SS . Following [59] we write ξS as
ξS = ξSr (r)S
S dr + ξSV (r)∇ i SS dyi. (4.49)
Our perturbations then transform as
δhSrr = ξ
S
r
f ′
f
+ 2ξS′r , δh
S
r = ξ
S
r − ξSV
g′
g
+ ξ
S′
V ,
δHST = 2ξ
S
V , δH
S
L = f g
′ξSr −
2
3
S(S + 2)ξ
S
V ,
δAS =
fΦ′
Φ
ξSr , δB
S =
ξ
S
V
g
δCS = ξS ′V −
g′
g
ξ
S
V , δD
S
I = 0, and δE
S
I = 0. (4.50)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Similarly, under U(1) transformations δASI → δASI + dχSI with gauge parameter
χ
S
I = χˆ
S
I (r)S
S we find
δhSrr = δh
S
r = δH
S
T = δH
S
L = δA
S = δBS = δCS = 0,
δDSI = χˆ
S′
I , and δE
S
I = χˆ
S
I . (4.51)
Just as we did for the S = 0 mode, we now substitute our ansatz into the Einstein–Maxwell
action (4.1) and expand to second order in the perturbations
qˆ ≡ {hSrr , hSr ,HSL ,HST ,AS ,BS ,CS ,DSI ,FSI }. (4.52)
The resulting action, S(2)[qˆ, qˆ′] depends on both qˆ and its first derivative qˆ′.10 Crucially, the
dependence in the angular coordinates drops out, as it should. Upon further inspection, one
notes that by performing further integrations by parts we can in fact write S(2) in a form that
depends only algebraically on hSrr (i.e., it does not depend on derivatives of h
S
rr ). The associated
boundary terms again either cancel with the Gibbons–Hawking–York term or with one of the
boundary counter-terms. This means we can easily integrate out hSrr from the action (though
this requires the Wick-rotation described in section 3) and find a reduced action S˜(2) that does
not depend on hSrr . Upon further scrutiny, one finds that, upon a couple of integration by parts,
S˜(2) does not depend on derivatives of hSr so again it can be integrated out. At this stage we are
left with a quadratic action Sˆ(2) that depends only on
{HSL ,HST ,AS ,BS ,CS ,DSI ,FSI } (4.53)
and their first derivatives. At this stage, we introduce gauge-invariant variables with respect to
the U(1). Under such gauge transformations the variables {HSL ,HST ,AS ,BS ,CS} are already
invariant. HoweverDSI and F
S
I transform non-trivially, so we define the invariant combination
f SI ≡ DSI − ES′I ⇒ DSI = f SI + ES′I . (4.54)
Using this definition, we find that Sˆ(2) depends only on
{HSL ,HST ,AS ,BS ,CS , f SI } (4.55)
and their first derivatives (as one would expect from the U(1) gauge invariance). Furthermore,
f SI decouples from all other variables and appears in Sˆ
(2) as
Sˆ(2) = Sˇ(2) + 4π2
S(S + 2)
S + 1
3∑
I=1
∫ +∞
−∞
dr
√
f g( f SI )
2, (4.56)
where Sˇ(2) depends only on
{HSL ,HST ,AS ,BS ,CS}. (4.57)
This means that f SI are non-dynamical in the sense that it enters the action algebraically and
contributes positively to Sˆ(2). We may thus focus our attention on Sˇ(2).
10 To bring the action to this form, we integrated by parts term proportional to HS ′′T and H
S ′′
L , with the boundary terms
canceling with the Gibbons–Hawking–York boundary term in (4.1).
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
At this stage we introduce gauge invariant variables with respect to the diffeomorphisms
(4.50). Since ξSr and ξ
S
V enter the gauge transformations forH
S
L and H
S
T algebraically, we can
use HSL and H
S
T to easily construct gauge invariant variables as follows. Define
QS1 = A
S − Φ
′
6rΦ
[
3HSL + S(S + 2)H
S
T
]
, (4.58a)
QS2 = B
S − H
S
T
2g
, (4.58b)
QS3 = C
S − H
S
′
T
2
+
g′
g
HST . (4.58c)
Using the transformations (4.50) it is relatively simple to see that δQS1 = δQ
S
2 = δQ
S
3 = 0.
Using this relations, we can solve for AS , BS and CS and insert those expressions into Sˇ(2).
After some integrations by parts, the terms with HSL and H
S
T drop out, as they should due to
gauge invariance. At this stage, Sˇ(2) is a function of
{QS1 ,QS2 ,QS3 ,QS′1 ,QS′2 }. (4.59)
Remarkably, and for reasons we do not fully understand, QS3 only enters the action alge-
braically (and with positive coefficient for the quadratic term). This allows us to perform the
Gaussian integral over QS3 and be left with an effective action for {QS1 ,QS2 } which we denote
by S(2)F .
It turns out to be beneficial to perform one final change of variable and write
QS1 =
2
LΦ
[
qS1 −
gλS
(
r + 4L2ΦΦ′
)
2r
(
gλS + 24L2Φ
2
)qS2
]
, and QS2 = −
1
2LΦ
qS2 . (4.60)
The final action then takes the following form
S(2)F =
∫ +∞
−∞
dr
g3/2
8
√
f
[
f (DqS)IKIJ(DqS)J + qSI VIJqSJ
]
, (4.61)
where I, J ∈ {1, 2}, (DqS)I = qS
′
I + εI
JqSJ , εI
J = εIJK
IJ ,
εIJ =
[
0 λ
0 0
]
, (4.62)
and
λ =
(S + 1)
32π2(λS − 3)L2r f
(
gλS + 24L2Φ
2
) [12L4rΦ fΦ′(g− 4L2Φ2)
− 4L2Φ2(3g2 + gλSL2 + 3gL2 + 3L2r20 + 3r40)+ gλSr20(L2 + r20)+ 96L6Φ4
]
.
(4.63)
Finally, the symmetric matrix V is given in appendix A.1 and the symmetric matrix K is more
easily expressed in terms of its inverse
21
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
(K−1)11 =
g(S + 1)
256π2r2(λS − 3)(gλS + 24L2Φ2)
[
2g2L2λS
(
Φ′ − 2rΦ
g
)2
+
+ gr2(λS − 3)λS + 16gΦ
2(g+ L2)(λS − 3)
f
(
1− 4L
4Φ2
g
(
g+ L2
))
]
,
(4.64a)
(K−1)22 =
(S + 1)
64π2λS(λS − 3)
(
λSg+ 24L2Φ
2
)
, (4.64b)
(K−1)12 = 0. (4.64c)
It is not hard to show that K is positive definite for r0/L > 31/4/21/2 ≈ 0.930605. First we
note that (K−1)22 is positive definite, second we note that is only the last term in (K−1)11
that is not positive definite. However, it is a simple exercise to show that 1− 4L4Φ2
g(g+L2) > 0 for
r0/L > 31/4/21/2. Since 31/4/21/2 < rmin0 /L, all large wormholes have positive definite K.
The non-existence of negativemodes can then be investigated by looking at the properties of
V. As the reader can see in appendix A.1, V is a rather complicated matrix, whose eigenvalues
we can only compute numerically as a function of r, for particular values of  and r0/L. This
allows us to exclude portions of the (r0/L, S) plane as potential regions with negative modes.
In figure 6 we plot the regions of parameter space where V is positive definite. It appears that
if V is positive definite for given (r0/L, S), then it remains positive definite if we increase S
while holding r0/L fixed. Since we are looking at large wormholes, we took r0/L > 1.
We see, perhaps counter-intuitively, that the larger the value of r0/L, the larger value of S
we have to achieve to seeV being positive definite. This might at first look worrying, but in fact
it is natural to expect angular momentum to have less effect at larger r0 (since the gradients
associated with given S are smaller at large r0). Indeed, as r0/L increases we find that the
most negative eigenvalue of V moves toward zero. Thus V becomes less and less negative at
large r0.
Note that ifV is not positive definite, it does not necessarily mean that negativemodes exist.
Indeed, for the complementary region we resort to solving for the spectrum numerically. After
doing so, we find no evidence for the existence of negativemodes for r0 > L and for S  2.We
have performed a thorough search in parameter space by scanning the large wormhole branch
to about r0 ∼ 104 and up to S ∼ 103.
4.3.3. Scalar-derived perturbations: the S = 1 sector. This mode is special, because HT no
longer appears in the metric perturbation. Apart from that, the construction is similar to what
we have done for S  2. Perhaps the end result is somehow surprising. Again, we find that
the second order action for the perturbations can be brought to a form similar to (4.56) with
again f 1I contributing positively to the action. However, we find that the second order action
for the remaining variables vanishes identically. Thus in the linearized theory these additional
variables are pure gauge. This in turn means that they are the linearization of a pure-gauge
mode in the full theory, as the method applied by [60] to showing that Einstein–Maxwell
theory admits a symmetric-hyperbolic formulation will also apply to our system. This in turn
means that no gauge degrees of freedom can remain in the linearized theory once the gauge
symmetries of the full theory have been fixed. We thus find that our wormhole is stable with
respect to gauge-invariant perturbations in this sector.
22
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 6. Disks represent regions of moduli space in the (r0/L, S) plane where V is
positive definite, and thus no negative mode exists.
4.3.4. Vector-derived perturbations: the V  2 sector. Things get more complicated, perhaps
unexpectedly, when we move on to study vector-derived or tensor-derived perturbations. Note
that when AI = 0, these sectors are really easy to study! The issue is that one can contract
the fundamental vector harmonics Si with AIi and build a scalar harmonic. So, in general, the
vectors harmonics couple to scalar-derived perturbations. Thus their treatment will require all
of the complications discussed above in the context of scalar-derived perturbations as well as
treatment of the vector harmonics.
An explicit discussion is thus extremely tedious but can be performed using precisely the
same techniques as in section 4.3.2.We suppress the details, but provide the following remarks.
It turns out that the vector derived perturbations only excite a few scalar-derived perturbations
and that they do not excite tensors-derived perturbations. For a given vector harmonic SVi
with V  2 we find that we need to consider a sum of three scalar harmonics of the form
SV−1 and three harmonics of the form SV+1. In each of these sectors, one of the harmonics
is proportional to cosψ, sinψ or has no ψ dependence. It is also possible to find the exact
differential map between these harmonics. It is then an incredibly tedious exercise to find the
resulting action, and diagonalise accordingly using appropriate numerics.We have done so and
find that the action is again positive definite.
However, there is a sector in which this unpleasant coupling does not occur and where one
can work purely with vector harmonics. We will describe the calculations for this simpler case
in detail to illustrate the mechanics of working with the vector harmonics themselves. The
simple sector involves vector-harmonics of the form
S
V
i dy
i = [− sin(mψ)dθ + cos(mψ) sin θ dϕˆ] sin|m−1| θ. (4.65)
23
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Regularity at θ = 0 and θ = π demands that m ∈ Z. The case with m = 1 is special, and
because the background is invariant under ψ →−ψ, φ→−φ we can take m  2 without loss
of generality. It is a simple exercise to show that V = 2m− 1.
For the metric perturbation we take
δ ds2V = 2h
V
r (r)S
V
i dr dy
i + HVT (r)S
V
i j dy
i dy j (4.66)
with
S
T
i j = ∇ iSVj +∇ jSVi . (4.67)
While for the gauge field perturbations we take
δAVI = A
V (r)£˜ΞV AI + B
S(r)AiIS
V
i dr + C
V
I (r)S
V
i dy
i, (4.68)
where £˜ is the Lie derivative acting on S3 and
ΞV = SiV
∂
∂yi
. (4.69)
Just as before we can ask how these perturbations behave under infinitesimal coordinate
transformations. Here, the infinitesimal diffeomorphisms are parameterized via
ξV = LV (r)SiV
∂
∂yi
, (4.70)
and it turns out this induces the following transformations
δhVr = L
V ′ − LV g
′
g
, δHVT = L
V , δAV =
LV
g
,
δBV = LV ′ − LV g
′
g
and δCV = 0. (4.71)
From here onwards, the procedures are very similar to what we have seen for the scalars.
First, we compute the second order action S(2) which is a function of hVr , h
V
T , A
V , BVr ,
CVr and their first derivatives. Furthermore, it also depends on the second derivatives of h
V
T .
One can integrate by parts the term proportional to the second derivative of hVT , reducing
S(2) to a function of first derivatives only. The resulting boundary term is canceled by the
Gibbons–Hawking–York boundary action, as it should.
One then notes that S(2) only depends on hVr , but not on its first derivative. This means we
can perform the (here already with the convergent sign) Gaussian integral over hVr and find an
action S˜(2) depending on hVT , A
V , BVr , C
V
r and their first derivatives. At this point we further
notice that CVr completely decouples from the remaining action, and furthermore that its con-
tribution to S˜(2) is manifestly positive definite. We are thus left to study an effective action Sˆ(2)
for hVT , A
V , BVr and their first derivatives. Finally, we make use of the gauge transformations
(4.71) and introduce gauge invariant variables of the form
qV1 = A
V − h
V
T
g
and qV2 = B
V +
g′
g
hVT − hV ′T . (4.72)
It is a simple exercise to check that δqV1 = δq
V
2 . After some integration by parts, the depen-
dence in hVT drops out completely (as it should by virtue of diffeomorphism invariance), and
24
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Sˆ(2) is solely written in terms of qV1 , its first derivative and of q
V
2 . Since q
V
2 only enters alge-
braically, we can perform the Gaussian path integral and find an action S(2)F for q
V
1 only. It is
convenient to perform one last change of variable and write
qV1 =
QV
Φ
. (4.73)
The final action for QV reads
S(2)F =
32π5/2L2(m− 1)3(m+ 1)Γ(m)
Γ
(
m+ 12
) ∫ +∞
−∞
dr
g3/2√
f
{
f
g(m− 1)m+ 4L2Φ2Q
V ′2
+
VV (r)
[g(m− 1)m+ 4L2Φ2]2Q
V
2
}
, (4.74)
with
VV (r) =
4
g2L2(m− 1)2
{
2 f gL4mΦrΦ′ + 4L2Φ2
[
L2
(
gm2 − 2gm+ r20
)
+ r40
]
+ gmL2
[
m
(
gm2 − gm+ r20
)
+ r20
]
+ m(m+ 1)gr40
}
. (4.75)
Form  2 all terms appearing inVV (r) are positive definite (note that rΦ′ is positive definite
for r ∈ R), and thus no negative mode exists in this sector as well.
4.3.5. Vector-derivedperturbations: the V = 1 sector. Thismode is again special because S
V
i j
vanishes identically, and thus hVT does not enter the calculation. Again, we find that C
1 con-
tributes positively to the action, but the remaining gauge invariant variables have a vanishing
action (and thus as in section 4.3.3 are in fact pure gauge under some special residual gauge
transformations). We thus find that our wormhole is stable with respect to gauge-invariant
perturbations in this sector.
4.3.6. Tensor-derived perturbations: the   2 sector. The analysis of general tensor-derived
perturbations is even more complicated than in the vector-derived case. The tensor-derived
perturbations in general couple to both scalar-derived (with S = T ± 2) and vector-derived
perturbations (with V = T ± 1). Once again, the general calculation can be performed by
combining an analysis of the tensor-derived harmonics with the techniques used above, and
doing so yields an action which (with appropriate numerics) can be verified to be positive
definite. As in the vector-derived case we suppress the details of this extremely tedious general
study and limit explicit discussion to a special type of tensor-derived perturbation that does not
source either vector- or scalar-derived harmonics. This allows us to illustrate the treatment of
the purely tensor-derived part. Combining such a treatment with the methods used above for
scalar- and vector-derived perturbations then suffices to treat the general case.
The simple tensor-derived sector is described by metric perturbations of the form
aδ ds2 =
g
2
√
2L2
hT(r)
{
(σ22 − σ21) cos [(m− 2)ψ]+ 2σ1σ2 sin [(m− 2)ψ]
}
sin|m−2| θ
≡ g
2
√
2L2
hT(r)Si j dyi dy j, (4.76)
25
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
with m ∈ Z and without loss of generality we take m  2. For the gauge field perturbation we
take
δAI =
g
8ΦL2
aT(r)Si j dy
i AI
j. (4.77)
Both hT (r) and aT (r) are automatically gauge invariant with respect to both infinitesimal dif-
feomorphisms and gauge perturbations. To show this, recall that Si j is transverse and trace
free. One can readily compute T by using equation (4.33c) and it turns out T = 2(m− 1).
Everything is much simpler now because these quantities are gauge invariant. In particular,
in order to cast the quadratic action in an adequate form we only need to remove a term
proportional to the second derivative of hT . The boundary term readily cancels off the usual
Gibbons–Hawking–York term. It turns out we can write the second order action in the form
S(2)F =
π5/2Γ(m− 1)
L4Γ
(
m− 12
) ∫ +∞
−∞
dr
g3/2
f 1/2
[
f (DqS)IKIJ(DqS)J + qSI VIJqSJ
]
, (4.78)
where I, J ∈ {1, 2}, (DqS)I = qS
′
I + εI
JqSJ , εI
J = εIJK
IJ ,
εIJ =
[
0 β1Φ
′
β2Φ
′ 0
]
, (4.79)
where β1, β2, β3 and β4 are constants,
K
IJ =
⎡
⎢⎢⎣
1+
L2
g
(β4 + β3Φ)2
L2
g
(β4 + β3Φ)2
L2
g
(β4 + β3Φ)2
L2
g
⎤
⎥⎥⎦ (4.80)
and
hT = q1 and aT = (β4 + β3Φ)q1 + q2. (4.81)
Note that det K > 0 andTr K > 0 so thatK is a positive definite symmetricmatrix. In addition,
we must take β3 = 2
√
2− β1 + β2 so that terms of the form q′IqJ for I 
= J do not feature the
action. The symmetric matrix V is incredibly cumbersome to write down explicitly, and not
very illuminating. What is important is that we can choose β1, β2 and β4 so that V is also
positive. A good choice turns out to be
β1 = −65
√
2(m− 1), β2 = −25
√
2(3m+ 2) and β4 =
17(m− 1)2
5
√
2(6m− 1) .
(4.82)
For the above choice, we present V in appendix A.2.
5. A scalar field model with S3 boundary
Let us now consider a different class of simple asymptotically AdS4 models in which the
wormhole is sourced by the stress-energy of scalar fields. We again assume spherical sym-
metry, and in particular a spherical boundary metric. After giving an overview of the model,
we briefly describe the ansätze we use to study the connected wormhole and the disconnected
26
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
solution. Computing the actions require more numerics than in the Einstein–Maxwell model
of section 4, so we devote a separate subsection to discussing the results of such computations.
A final subsection considers potential negative modes in direct parallel with the discussion
of section 4.3. The final results also mirror those of section 4, as we again find a Hawk-
ing–Page-like structure with two branches of wormholes (large and small). Moreover, the large
wormholes are once again free of negative modes and dominate over the disconnected solution
when they are sufficiently large. However, this model also has no possible notion of a brane-
nucleation negative modes as the theory does not contain branes. In particular, while we will
later discuss its relation to certain string-theoretic setups, the currentmodel is notUV-complete.
Wewill study both conformally coupled scalars andmassless scalars, though for themoment
we include an arbitrary mass parameter μ. Our action is given by
S = −
∫
M
d4x
√
g
[
R+
6
L2
− 2(∇aΠ) · (∇aΠ)∗ − 2μ2Π · Π∗
]
− 2
∫
∂M
d3x
√
hK + Sμ
2
B , (5.1)
where L is the four-dimensional AdS length scale, Π describes a doublet of complex scalar
fields, and ∗ denotes complex conjugation. The second term in (5.1) is the usual Gib-
bons–Hawking term and Sμ
2
B the boundary counter-term to make the action finite and the
variational problem well defined. The precise form of Sμ
2
B will explicitly depend on μ
2.
The Einstein equation and scalar field equation derived from this action read
Rab − R2 gab −
3
L2
gab = 2∇(aΠ · ∇b)Π∗ − gab∇cΠ · ∇cΠ∗ − μ2gabΠ · Π∗,
(5.2a)
Π = μ2Π. (5.2b)
We now note that if we take the trace of the Einstein equation, we find
R = −12
L2
+ 2(∇aΠ) · (∇aΠ)+ 4μ2Π · Π∗. (5.3)
As such, the on-shell Euclidean action can be computed by evaluating the following bulk
integral
Son−shell =
∫
M
d4x
√
g
[
6
L2
− 2μ2Π · Π∗
]
− 2
∫
∂M
d3x
√
hK + Sμ
2
B . (5.4)
The precise form of SB will be important to evaluate this on-shell action and depends on the
boundary conditions that we impose on scalar doublet Π. We will take μ2 to be zero or to take
the conformal value μ2L2 = −2.
In standard Fefferman–Graham coordinates [54] the metric takes schematic form
ds2 =
L2
z2
[
dz2 + gˆμν(x, z)dxμ dxν
]
, (5.5)
where z = 0 marks the location of the conformal boundary and recall that Greek indices run
over boundary directions only. One can then show that gˆμν(x, z) admits a simple expansion in
terms of a power series in z, possibly with log z terms depending on the scalar field mass μ.
27
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
For μ2L2 = −2 and μ = 0 the log z terms can be shown to be absent and gˆμν(x, z) can be
expanded as
gˆμν(x, z) = g0μν(x)+ z
2g(2)μν(x)+ z
3g(3)μν(x)+ o(z
3), (5.6)
where g0μν(x) is interpreted as the boundary metric. We can now explain a little bit better with
what we mean by ∂M in equation (5.1). The surface ∂M is defined as the ε→ 0 limit of
hypersurfaces ∂Mε on which z = ε. Furthermore, hμν(ε) is the induced metric on ∂Mε. In
this sense, limε→0 ε2hμν/L2 = g0μν(x). Note that in the wormhole case ∂M has two connected
components, one at each end of the wormhole.
In FG coordinates the scalar doublet Π can be expanded as
Π = Π−(x)zΔ−[1+ o(1)]+ Π+(x)zΔ+ [1+ o(1)], (5.7)
with
Δ± =
3
2
±
√
9
4
+ μ2L2. (5.8)
Throughout this section fix Π+ as a boundary condition (associated with ‘standard
quantization’ in the language of [61]), so that in AdS/CFT Π−(x) becomes the expectation
value of the operator dual to Π. For the massless and conformal cases we have Δ− = 0 and
Δ− = 1 respectively. This choice (partially) dictates what SB should be to make the variational
problem well defined. That is to say, when deriving the equations of motion we want to ensure
that we keep Π−(x) fixed and that our boundary terms are consistent with such choice. The
remaining freedom in choosing SB is fixed so that the action (5.1) is finite.
In the massless case we take
Sμ
2=0
B = −
4
L
∫
∂M
d3x
√
h− L
∫
∂M
d3x
√
hRh + 2L
∫
∂M
d3x
√
hhμν∇hμΦ · ∇hν Φ,
(5.9)
where Rh is the Ricci scalar associated with h and∇h its metric preserving connection.
For the conformal case we take
Sμ
2L2=−2
B = −
4
L
∫
∂M
d3x
√
h− L
∫
∂M
d3x
√
hRh − 2
L
∫
∂M
d3x
√
hΦ · Φ. (5.10)
We are interested in finding solutions where the metric enjoys spherical symmetry but the
scalars are chosen in such a way that these symmetries are broken. In particular, we might
consider
ds2 = grr(r)dr2 + gS3 (r)dΩ
2
3, (5.11)
where dΩ23 is the unit round three-sphere, grr(r), gS3 (r) are to be determined later and r is an
arbitrary bulk coordinate.
For the scalar fields, we take
Π = XΠ(r) (5.12)
withΠ(r) ∈ R and X a two dimensional complex unit vector on S3 with dX 
= 0 and X · X∗ = 1.
In fact, the coordinates of (5.11) turn out to be inconvenient for constructing our solutions.
We thus present two new ansätze below corresponding to connected or disconnected solutions
and which specify our gauge choice slightly differently in each case.
28
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
5.1. Ansatz for the wormhole solutions
When searching for wormhole solutions, we will take
ds2 =
L2
(1− y˜2)2
{
4 f (y˜)dy˜2
2− y˜2 + y
2
0 dΩ
2
3
}
. (5.13)
Clearly, equation (5.13) falls into the same symmetry class as equation (5.11), and the factors
of y˜ ∈ (−1, 1) where chosen to that asymptotically (as y˜→±1) f → 1. From the form of the
ansatz above it is clear that y0 is the minimal radius of the S
3 in the interior, which is attained
at y˜ = 0. The value of this radius for given boundary sources will be determined numerically.
For the scalar field we take
Π = (1− y˜2)Δ+q. (5.14)
Let us describe the boundary conditions at y˜ = 0 in detail. We wish to impose the reflection
symmetry y˜→−y˜which leaves the locus y˜ = 0 invariant.As a result, f (y˜) = f (−y˜) and q(y˜) =
q(−y˜). This implies
d f
dy˜
∣∣∣∣
y˜=0
=
dq
dy˜
∣∣∣∣
y˜=0
= 0. (5.15)
The requirement (5.15) in fact motivates us to choose a different coordinate that automatically
enforces these conditions. In particular, we use y := y˜2 with y ∈ (0, 1). The line element now
reads
ds2 =
L2
(1− y)2
{
f (y)dy2
(2− y)y + y
2
0 dΩ
2
3
}
, (5.16)
and the boundary conditions (5.15) become just the statement that f and q are regular at y = 0
in the sense that (using further input from the 2nd order equations of motion) both must admit
a Taylor series expansion about this locus. The equations in the y coordinates and in this gauge
are sufficiently compact to present here:
(1− y)4√2− y√y√
f
d
dy
[ √
2− y√y√
f (1− y)2
dΠ
dy
]
− 3(1− y)
2
y20
Π− L2μ2Π = 0,
(5.17a)
f =
(2− y)yy20
3
[
(1− y)2 + y20
]−Π2 [3(1− y)2 + μ2L2y20]
[
3− (1− y)2
(
dΠ
dy
)2]
.
(5.17b)
A non-singular wormhole must have f being non-zero and finite everywhere, and in particular
at y = 0. Looking at the expression above for f , it follows that at y = 0 we must have
Π(0) =
√
3
√
1+ y20√
3+ μ2L2y20
, (5.18a)
29
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
where without loss of generality we took q(0) > 0. Assuming a regular Taylor series for q
around y = 0, it also follows that
dΠ
dy
∣∣∣∣
y=0
=
√
3
√
1+ y20
(
3+ L2y20μ
2
)3/2
y20
(
3− L2μ2) and f (0) = 3+ L
2y20μ
2
3− L2μ2 . (5.18b)
Note that the four-dimensional Breitenlohner–Freedman bound ensures that both quantities
above are positive definite.
At the conformal boundary we find
q =
⎧⎪⎪⎨
⎪⎪⎩
V − 3
2y20
V(1− y)2 + λ(1− y)3 − 9
8y40
V(1− y)4 +O [(1− y)5] for μ = 0,
V
y0
+ κ(1− y)+ V
(
4+ V2
)
2y30
(1− y)2 +O [(1− y)3] for μ2L2 = −2,
(5.19)
where λ and κ are constants. The metric (5.16) is not yet in FG coordinates, so we are not yet
ready to read off the values of boundary sources. In order to achieve this we need to perform a
change of variables from y to z. We only require this change asymptotically,
y =
⎧⎪⎨
⎪⎩
1− y0z− 14
(
1− V2) y0z3 +O(z5) for μ = 0,
1− y0z− 14
(
1+ V2
)
y0z
3 − 4
9
κVy30z
4 +O(z5) for μL2 = −2.
(5.20)
We can now compare the expansion for Π in powers of z with the general expansion (5.7)
which gives
Π+ = XV (5.21)
for both μ = 0 and μ2L2 = −2. We thus see that V is to be interpreted as the source of the
operator dual to Π.
The numerical procedure to find these solutions is clear. We take a value for y0, and use
the boundary conditions (5.1) to numerically integrate the equations outwards to y = 1. Once
this is done, we read off V , thus finding what source was needed to source a wormhole with
minimal size y0. To do this, we write the equations in first order form, and use a Chebyshev
collocation grid on Gauss–Lobatto points to perform the numerical integration. Because in our
gauge the expansion of q can be shown to be analytic at the singular points y = 0, 1 we obtain
exponential convergence in the number of grid points as we approach the continuum limit.
Our integration also determines the values of λ and κ in (5.19). These are related to the
expectation value 〈OΠ〉 of the operator dual to Π via
〈 O Π〉 =
δS
δΦ+
= X
{−24π2λL2y30 for μ = 0,
8π2κL2y20 for μL
2 = −2.
(5.22)
Finally, we give a few words on how to evaluate (5.4) in a numerically stable manner. We
first look at how the first term in (5.4) diverges in inverse powers of (1− y) and √y. We then
30
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
add and subtract a regulator with precisely the same singularity structure, but one that we can
integrate analytically. More precisely we use
S
8π2L2
=
∫ +∞
0
dy
√
f y30
(1− y)4√2− y√y
(
3− L2μ2Π2)− 2∫
∂M
d3x
√
hK + Sμ
2
B
=
∫ +∞
0
dy
[ √
f y30
(1− y)4√2− y√y
(
3− L2μ2Π2)− G]
+
∫ +∞
0
dyG− 2
∫
∂M
d3x
√
hK + Sμ
2
B , (5.23)
where G takes the form
G =
G(−4)
(1− y)4 +
G(−2)
(1− y)2 + G
(0) +
G(1)(1− y)√
y
. (5.24)
Here allG(i) are constant and are chosen in such a way that the first integrand on the second line
of (5.23) vanishes as y→ 1 and as y→ 0. These constants can be found analytically, becausewe
know the expansion for all functions via (5.19). Once this is done, we can analytically perform
the integrations on the last line of (5.23) and check that the result is finite as desired. The
numerical integral that remains is then manifestly finite, with all the complicated cancelations
having been implemented analytically.
5.2. Ansatz for the disconnected solution
The procedure for the disconnected solutions is similar to what we have just described for the
wormhole, so we will be more brief here. The ansatz reads
ds2 =
L2
(1− y˜2)2
{
4 f (y˜)dy˜2
2− y˜2 + y˜
2(2− y˜2)dΩ23
}
, (5.25)
with the regular center located at y˜ = 0 and the conformal boundary at y˜ = 1. Regularity at
y˜ = 0 now demands that
f (y) = 1+O(y˜2) and Π(y) = Q(0)y˜[1+O(y˜2)], (5.26)
with Q(0) constant. These regularity conditions suggest introducing a coordinate y˜2 = y just as
before, and they also suggest redefining
Π =
√
y
√
2− y(1− y)Δ+q. (5.27)
At the conformal boundary y = 1 we demand Π(1) = q(1) = V . For both μ = 0 and
μ2L2 = −2 this coincides with fixing the source for the expectation value dual to Π.
Just as before, the Einstein equation and scalar field equation yield a single equation for
q which we can solve numerically given the boundary conditions above. The procedures to
regulate the action and to identify the source are very similar to the ones used for the wormhole
geometry, so we will suppress this discussion here and pass directly to results below.
5.3. Results
Let us define
ΔS = 2SD − SW , (5.28)
31
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 7. Action differenceΔS as a function of the source V for μ = 0. There is a Hawk-
ing–Page transition around V > VHP ≈ 3.70655(6). The large wormholes are depicted
as blue disks and the small wormholes as orange squares.
where SD is the Euclidean on-shell action for the disconnected solution and SW the on-shell
action for the wormhole solution with the same values of the boundary sources Π+. Note that
both of these actions are finite due to the counter-terms.
We first focus on the massless case μ = 0. As shown in figure 7, our numerical results
indicate that ΔS crosses zero and becomes positive for V > VHP ≈ 3.70655(6). In addition,
for any V  Vmin ≈ 3.38021(4), there are two wormholes for a given value of V . At Vmin we
find y0 = ymin0 ≈ 2.13813(6). The solution with larger ΔS we will call the large wormhole,
and the one with lower ΔS we call the small wormhole. These phases are depicted as blue
disks (large wormhole) and orange squares (small wormhole), respectively, in figure 7. For the
small wormhole phase we find ΔS < 0 in all cases. To backup our nomenclature, we plot the
minimum radius y0 of the S
3 as a function of V in figure 8 using the same color coding as in
figure 7.
Although the conformally coupled case is qualitatively similar, it turns out to be more chal-
lenging numerically. This is because in that case V needs to be very large for the wormholes
to exist, and even larger for the large wormholes to dominate (i.e., to see the Hawking–Page-
like transition). We find the transition to occur at SD ∼ 5× 109L2 and SW ∼ 1010L2, while the
difference is of order ΔS ∼ L2. This means we have to accurately extract the first 15 digits
when evaluating the regulated integrals to determine the transition with good accuracy. At this
point the use of high-precision arithmeticswas essential.We used octuple precision throughout,
keeping track of at least the first 256 digits.
The resulting phase diagram is similar in structure to what we found in massless case. For
V  Vmin ≈ 179.55054(5)we find twowormhole solutions for a given value ofV . At this point,
y0 = ymin0 ≈ 0.8416(7). TheHawking–Page transition occurs on the largewormhole branch for
32
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 8. Minimum value of the S3 radius as a function of V for μ = 0. The blue disks
represent the large wormholes and the orange disks the small wormholes. Wormholes
only exist for V > Vmin ≈ 3.38021(4).
V = VHP ≈ 639.20819(5); see figure 9. The two wormhole solutions for each V can again be
distinguished by the different values of y0; see figure 10 where we used the same color coding
as in figure 7.
5.4. Negative modes
We now discuss perturbations around our scalar wormholes, and in particular, the potential
existence of negative modes. Just as we did for the U(1)3-Maxwell wormholes, we will take
advantage of the SO(4) symmetry of the S3 to decompose the perturbations into the spherical
harmonics (4.33).
5.4.1. The scalar homogeneous mode: S = 0. The metric perturbations are given by
δ ds2 =
L2
(1− y)2
[
δ f (y)dy2
(2− y)y + δg(y)dΩ
2
3
]
, (5.29a)
and
δΠ = XδΠ(y). (5.29b)
Throughout, we shall work with gauge invariant variables. The most general infinitesimal dif-
feomorphism compatible with SO(4) takes the form ξ = L2ξy(y)dy. This in turn induces the
33
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 9. Action difference ΔS as a function of the source V for μ2L2 = −2. There is
a Hawking–Page transition around V > VHP ≈ 639.20819(5). The large wormholes are
depicted as blue disks and the small wormholes as orange squares.
following gauge transformations on the different metric and scalar perturbations
δ f =
[
2(1− y) (2y2 − 4y+ 1)− (1− y)2(2− y)y f ′
f
]
ξy + 2(2− y)y(1− y)2ξ′y,
(5.30a)
δg =
2(1− y)(2− y)y
f
ξy, (5.30b)
δΠ =
(1− y)2(2− y)yΠ′
f
ξy, (5.30c)
where ′ denotes differentiation with respect to y.
Our procedure will be similar to the one we used for the U(1)3-Maxwell wormhole in
section 4.We first evaluate the action (5.1) to second order in {δ f , δg, δΠ} and their derivatives.
Let us denote this quadratic action by S(2). Second derivatives of δg with respect to y appear in
the action, which we integrate by parts to write the action in first order form. This procedure
generates a boundary term which cancels the perturbed Gibbons–Hawking–York boundary
term. The resulting action S(2) is written in terms of {δ f , δg, δΠ} and their first derivatives
only. We then note that, after some integration by parts whose surface term vanishes or can-
cels with the boundary counterterms, δ f only enters the action algebraically. That is to say, no
derivatives act on δ f . This means we can formally perform the Gaussian integral over δ f , and
find a new action S˜(2) that depends only on {δg, δΠ} and their first derivatives.
34
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 10. Minimum value of the S3 radius as a function of V for μ2L2 = −2. The
blue disks represent the large wormholes and the orange disks the small wormholes.
Wormholes only exist for V > Vmin ≈ 179.55054(5).
At this stage we introduce gauge invariant quantities. Looking at the transformations (5.30),
we introduce a gauge invariant variable Q through the relation
δΠ = Q+
1
2
(1− y)q′δg. (5.31)
Substituting δΠ into S˜(2) yields an action for Q and its first derivative Q′. To reduce the
action to this form, more integrations by parts have to be performed and, remarkably, every
non-vanishing term at the wormhole boundaries cancel with contributions from the perturbed
boundary counter-terms. It turns out that the action takes a simpler form if we further redefine
Q =
√
3− (1− y)2Π′2√
6
Q˜. (5.32)
The second order action S˜(2) then takes the following rather explicit form
S˜(2) = 4π2L2
∫ +∞
0
dy
√
f y30
(1− y)4√2− y√y
[
(1− y)2(2− y)y
f
Q˜′2 + VQ˜2
]
,
(5.33)
where
35
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
V(y) =
1
Π2
{
9 f
[
Π
(
1−Π2)m3yq′ + 3 (m2y + y20)]2
(2− y)yy40z2y
+
3(2− y)y
f
[(
1+ myΠΠ′
)2
+ 1
]
− 8Π
2
(
1− q2)m4y
y20zy
[
3
(
6−Π2)
8Π
(
1−Π2)my +Π′
]2
− 18myΠΠ
′
zy
+
9
(
6−Π2)2m2y
8
(
1−Π2) y20zy −
45
(
m2y + y
2
0
)
y20zy
}
, (5.34)
with
my = 1− y and zy = 3− (1− y)2Π′2. (5.35)
The change of variable (5.32) onlymakes sense so long as the argument inside the square root is
positive. One can show analytically that, so long as y0 > yc0 ≈ 1.94712(2) for the massless case
and y0 > yc0 ≈ 0.77845(0) for the conformally coupled case, the argument inside the square
root is indeed positive definite. In both cases yc0 < y
min
0 , meaning that this transformationmakes
sense for all largewormholes, and for a range y0 ∈ (yc0, ymin0 ) of small wormholes.Recall that we
want to establish that the large wormhole branch has no negative modes so, for our purposes,
it suffices to study wormholes in the range y0 > yc0.
To search for negative modes we simply study the eigenvalue equation
− (1− y)
4√2− y√y√
f
[ √
2− y√y
(1− y)2√ f Q˜
′
]′
+ VQ˜ = λQ˜. (5.36)
Note that in the y˜ coordinates the potential is an even function of y˜ as expected from the Z2
symmetry. This means that we can study separately those eigenfunctions which are even and
odd in y˜. In terms of the y coordinates, this maps to functions that scale as Q˜ ∼ √y(a0 + b0y)
near y = 0 for the odd case while for the even case we have Q˜ ∼ a˜0 + b˜0y (where a0, b0, a˜0
and b˜0 are constants).
Before proceeding, we must specify the boundary conditions at infinity. A Frobenius
analysis near the conformal boundary reveals two possible near boundary behaviors,
Q˜ ∼ (1− y) 32±
√
9
4+μ
2L2−λ. (5.37)
The integrations by parts we have performed throughout our analysis yield boundary terms
that vanish only consistent if we choose the+ root. This motivates performing one last change
of variable,
Q˜ = (1− y) 32+
√
9
4+μ
2L2−λyε/2(2− y)ε/2q˜, (5.38)
where we set ε = 0 for the even sector and ε = 1 for the odd sector of perturbations.
We find no odd negative modes, and a single even negative mode exists in the regime
y0 ∈ (yc0, ymin0 ) but not for other values of y0. In particular we find no negative modes for the
large wormhole branch. Perhaps even more interesting, the negative mode of the small worm-
hole branch disappears precisely at y0 = ymin0 . This can be seen in figure 11 where we plot the
negative eigenvalue λ as a function of y0/ymin for the massless (left panel) and conformally
coupled scalars (right panel).
36
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 11. We find a negative mode on the small wormhole branch, which vanishes at
y0 = ymin0 . On the left panel we plot the negative mode for the massless case and on the
right the negative mode of the conformally coupled case. In both cases, the horizontal
axis is given by y0/ymin0 .
5.4.2. Scalar derivedmodeswith S  2. We now study scalar-derived perturbations in detail
for S  2. This turns out to be a much easier task than studying perturbations of the U(1)3
Maxwell theory with a spherical boundary metric. For the metric perturbations we take the
same ansatz as in equation (4.44)
δ ds2S = h
S
yy(y)S
S dy2 + 2hSy (y)∇ iSS dy dyi
+ HST (y)S
S
i j dy
i dy j + HSL (y)S
S𝕘i j dyi dy j, (5.39a)
while for the scalar perturbation we choose
δΠS =
XBS(y)S
S + (∇ iSS∇ iX)AS(y). (5.39b)
Under an infinitesimal diffeomorphism of the form
ξS = ξSy (y)dy+ L
S
y (y)∇ iSS dyi, (5.40)
the metric and scalar perturbations transform as
δhSyy =
(
1
y
− 1
2− y −
2
1− y −
f ′
f
)
ξSy + 2ξ
S
y
′
, (5.41a)
δhSy = −
2LSy
1− y + ξ
S
y + L
S
y
′
, (5.41b)
δHST = 2L
S
y , (5.41c)
δHSL = −
2
3
λSL
S
y +
2(2− y)yy20
(1− y) f ξ
S
y , (5.41d)
37
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
δAS =
(1− y)2Π
L2y20
LSy , (5.41e)
δBS =
(2− y)(1− y)2yΠ′
L2 f
ξSy . (5.41f)
Our procedure will again be very similar to the one we used for the U(1)3 Maxwell theory.
We first expand the action (5.1) to quadratic order in the perturbations, using S(2) to denote
the result. This quadratic action involves second derivatives of HST and H
S
L , which we readily
remove via an integration by parts. The resulting boundary term cancels the perturbed Gib-
bons–Hawking–York boundary term. At this stage, S(2) is a function of hSyy, h
S
y , H
S
T , H
S
L , A
S ,
BS and their first derivativeswith respect to y. However, after a few integration by parts, whose
boundary terms partially cancel some of the counter-terms, we can write S(2) in a way where
hSyy only enters the action algebraically. As such, it is easy to perform the (correctly-signed)
Gaussian integral over this variable and obtain a new action S˜(2) which is a function of hSy ,H
S
T ,
HSL , A
S , BS and their first derivatives. Upon a few more integration by parts, we can rewrite
S˜(2) in a manner where hSy again only enters algebraically. Since it also has the correct sign,
we can again perform the Gaussian integral to finally obtain an action Sˇ(2) which is a function
of HST , H
S
L , A
S , BS and their first derivatives.
At this stage we introduce two gauge invariant quantities built using HST , H
S
L , A
S , BS .
These are
QS1 = A
S − (1− y)
2Π
2L2y20
HST , and (5.42a)
QS2 = B
S − (1− y)
3Π′
2L2y20
(
HSL +
1
3
HST
)
. (5.42b)
We then solve for AS and BS in terms ofQS1 ,Q
S
2 ,H
S
L ,H
S
T , and substitute the resulting expres-
sion into Sˇ(2). After some integration by parts (which generate boundary terms that again cancel
some of the perturbed counter-terms) we find that all dependence on HSL and H
S
T disappears
(as it should due to gauge invariance). At this stage Sˇ(2) is a function of QS1 , Q
S
2 and their first
derivatives only.
To proceed, we now treat the massless and conformal cases separately. Strictly speaking,
the procedure we will apply to the conformally coupled case also works for the massless case,
but as we shall see it is much more cumbersome so we will be more schematic there. For the
massless case we take
QS1 ≡
(1+ λS)1/4
8
√
2π
√
λS
ΠqS1 ,
QS2 ≡
(1+ λS)1/4
8
√
2π
√
λS
(1− y)Π′qS2 . (5.43)
The reason why this change of variable is only adequate for the massless case is the presence of
the multiplying factor Π′ in the definition of qS2 . It is a relatively simple exercise to show that
Π must have an extremum for y ∈ (0, 1) for any μ2L2 < 0. This means that this redefinition
will necessarily include a singularity in those cases and is thus not appropriate to use. For the
massless case, however,Π is monotonic and as such Π′ does not vanish in y ∈ (0, 1).
38
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
For the massless case, the resulting quadratic action takes the form
Sˇ(2) = L2
∫ +∞
0
dy
√
f y30
(1− y)4√2− y√y
[
(1− y)2(2− y)y
f
qSI
′
K
IJqSJ
′
+ qSI V
IJqSJ
]
,
(5.44)
where
K
−1 =
1
λS − 3
⎡
⎣3Π2 + λS − 3 λSΠ2
λSΠ
2 λSΠ
2
[
λS − 3+ (1− y)2Π′2
]
(1− y)2Π′2
⎤
⎦ . (5.45)
Note that λS = S(S + 2), and we are focusing on S  2, so that λS  8. From the above
expression it is clear that Tr(K−1) > 0. Furthermore, we have
det(K−1) =
λSΠ
2
(1− y)2 (λS − 3)Π′2
[
λS − 3+ 3Π2 + (1− y)2
(
1−Π2)Π′2] .
(5.46)
It turns out that det(K−1) is not positive definite for all values of y0, but one can check numer-
ically that for S = 2 it is positive so long as y0 1.6859(2) < ymin0 . We thus conclude that
det(K−1) is positive for all large wormholes, and thus that K is positive definite for all large
wormholes. Note that for larger values of S, these values will become even smaller.
We now turn out attention to V, which has a rather complicated expression. However, it
turns out that the combination
VIJ = (K
−1)IK(K−1)JLVKL (5.47a)
has a more manageable form, namely
V11 =
m2y
y20 (λS − 3)Π2
(
λS − 3+ 3Π2
) (
λS − 4+ 4Π2
)
, (5.47b)
V12 = − myy20(λS − 3)ΠΠ′
λS
[
2 (λS − 3)− myΠ
(
λS − 4+ 4Π2
)
Π′
]
, (5.47c)
V22 =
λS
(2− y)myyy40 (λS − 3)Π′3
{
(2− y)myyy20λS
(
λS − 4+ 4Π2
)
Π′ + 6y20 (λS − 3)Π f
+ my f
[
3y20 (λS − 2)+ m2yλS −
(
6y20λS − 6y20 + 5m2yλS
)
Π2 + 4m2yλSΠ
4
]
Π′
}
,
(5.47d)
where one should recall that my = 1− y.
It is now a simple exercise to compute the two real eigenvalues λ−  λ+ of V as a function
of y. We then take the minimum value of λ− in the interval y ∈ (0, 1), and plot it as a function
of y0. It turns out a critical value of y0 exists above which λ− is positive definite for all λS  2.
For the massless case this occurs for y0 1.7435(6) < ymin0 . This establishes that no negative
modes exist in the scalar sector with S  2 for large wormholes generated by massless scalar
sources.
39
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
The conformally coupled case is more complicated becausewe cannot apply (5.43). Instead,
we have to use a procedure more similar to the one we used for the U(1)3 theory. Here we
consider a generic value of μ2L2 < 0 parametrized by the conformal dimensionΔ satisfying
μ2L2 = Δ(Δ− 3). (5.48)
Instead of (5.43) we consider
QS1 ≡
(1+ λS)1/4
8
√
2π
√
λS
(1− y)Δ
[
qS1 +
(1− y)ΠΠ′
λS − 3+ (1− y)2Π′2 q
S
2
]
,
QS2 ≡
(1+ λS)1/4
8
√
2π
√
λS
(1− y)ΔqS2 . (5.49)
The second order action now reads
Sˇ(2) = L2
∫ +∞
0
dy
√
f y30
(1− y)4√2− y√y
[
(1− y)2(2− y)y
f
(DqS)IKIJ(DqS)J + qSI VIJqSJ
]
,
(5.50)
with
K
−1 =
1
(1− y)2Δ
⎡
⎢⎢⎣
1+
Π2
[
3− (1− y)2Π′2]
λS − 3+ (1− y)2Π′2
0
0 λS
[
1+
(1− y)2Π′2
λS − 3
]
⎤
⎥⎥⎦ .
(5.51)
This result is positive so long as y0 0.694251 < ymin0 , thus again implying that it is positive
definite on the large wormhole branch. Furthermore,
(DqS)I = qSI
′
+ εI
JqSJ , (5.52)
where εI J = εIKKKJ and
εIK =
{(
3m2y + L
2y20μ
2
)
Π− my
[
2m2y + 3y
2
0 −
(
2m2y + L
2y20μ
2
)
Π2
]
Π′
}
× m
−1−2Δ
y λSΠ f
y20y(2− y)
(
λS − 3+ m2yΠ′2
) [ 0 1−1 0
]
. (5.53)
ThematrixV turns out to be positive definite so long as y0 0.75186(7) < ymin0 , thus rendering
the large wormholes branch stable. The expression forV can be found in appendix B. The sec-
ond formalism described above can also be used to study the massless case, but the expressions
are considerably more complicated than in the approach described for m = 0 above.
5.4.3. Scalar-derived perturbations with S = 1. As we have seen, scalar modes with S = 1
are excluded from the previous analysis due to the fact that S1i j = 0 for this special mode. Our
metric perturbation becomes simpler in this case where it takes the form
δ ds2S=1 = h
1
yy(y)S
1 dy2 + 2h1y(y)∇ iS1 dy dyi + H1L(y)S1𝕘i j dyi dy j, (5.54a)
40
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
while the scalar perturbation is essentially unchanged and yields
δΠ = XB1(y)S1 + (∇ iS1∇ iX)A1(y). (5.54b)
Since scalar derived infinitesimal diffeomorphisms still have two degrees of freedom,
ξS=1 = ξ1y (y)dy+ L
1
y(y)∇ iS1 dyi, (5.55)
we only expect a single gauge-invariant master function. We shall see that this is indeed the
case. Under such diffeomorphisms the metric and scalar perturbation functions transform as
δh1yy =
(
1
y
− 1
2− y −
2
1− y −
f ′
f
)
ξ1y + 2ξ
1
y
′
, (5.56a)
δh1y = −
2LSy
1− y + ξ
1
y + L
1
y
′
, (5.56b)
δH1L = −2L1y +
2(2− y)yy20
(1− y) f ξ
1
y , (5.56c)
δA1 =
(1− y)2Π
L2y20
L1y , (5.56d)
δB1 =
(2− y)(1− y)2yΠ′
L2 f
ξ1y . (5.56e)
Just as for the case with S  2, both hSyy and hSy can be integrated out, leaving an action which
depends only on H1L, A
1
L and B
1
L. At this stage we introduce a gauge invariant variable
Q1 = ΠB1 − (1− y)Π′A1 − (1− y)
3ΠΠ′
2L2y20
H1L. (5.57)
The resulting quadratic action Sˇ(2) can be written entirely in terms ofQ1 and its first derivatives
and takes the form
Sˇ(2)
12π2
=
∫ +∞
0
dy
√
f y30
(1− y)4√2− y√y
[
(2− y)y
f
m2yΠ
4
3Π2 + m2y
(
1−Π2)Π′2Q1′2 + V1 Q12
]
,
(5.58)
with
V1 =
Π2[
3Π2 + m2y
(
1−Π2)Π′2]2y20
{
36
(
m2y + y
2
0
)−Π [36 (m2y + y20)Π− 9m2yΠ3
− 6my
(
7m2y + 6y
2
0 − m2yΠ2
)
Π′ − 3m2yΠ
(
5m2y + 2y
2
0 − 3m2yΠ2
)
Π′2
− 2m5y
(
1−Π2)2Π′3]− 2(2− y)yy20
f
[
9− Π (9Π− 18myΠ′ − 12m2yΠΠ′2
+ 3m3yΠ
2Π′3 + 3m4yΠΠ
′4 + 2m5yΠ
′5 − m5yΠ2Π′5
)]
− 6
(
m2y + y
2
0
)
f
(2− y)yy20
(
1−Π2) [3 (m2y + y20)+ m3yΠ (1−Π2)Π′]
}
. (5.59)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Once more, one can easily verify that V1 and 3Π2 + m2y
(
1−Π2)Π′2 are positive definite
along large wormhole branch. This thus establishes that S = 1 scalar derived perturbations
yield no negative modes on the large wormhole branch.
5.4.4. Vector-derived perturbations with V  2. This sector of perturbations turns out to be
much easier to study than in the U(1)3−Maxwell theory. This is because in the current case
the vector-derived perturbations do not source scalar-derived perturbations. Our ansatz for the
metric perturbations reads
δ ds2V = 2h
V
y (y)S
V
i dy dy
i + HVT (y)S
V
i j dy
i dy j, (5.60a)
while for the scalar perturbation we choose
δΠV = (S
V
i ∇ iX)AV (y). (5.60b)
The most general vector-derived infinitesimal diffeomorphism can be written
ξV = LVy S
V
i dy
i. (5.61)
This infinitesimal diffeomorphism induces the gauge transformation
δhVy = −
2LVy
1− y + L
V
y
′
, (5.62a)
δHVT = L
V
y , (5.62b)
δAV =
(1− y)2Π
L2y20
LVy . (5.62c)
The procedurenow is very similar to whatwe have seen for the scalar-derivedperturbations.We
first derive the second order action S(2) and note that HVT appears in the action with terms that
involve second derivativeswith respect to y.We integrate these by parts, with the non-vanishing
boundary terms canceling the perturbed Gibbons–Hawking–York term. Furthermore, after
some further integration by parts, hVy only enters the action algebraically. This means we can
perform the Gaussian path integral and find a new second order action Sˇ(2) which depends on
HVT , A
V and their first derivatives with respect to y. At this point, we introduce the gauge
invariant variable
QV = AV − (1− y)
2Π
L2y20
HVT . (5.63)
Solving this expression with respect to AV and substituting it back into Sˇ(2) gives an action for
QV and its first derivative with respect to y. The dependence inHVT completely drops out from
the calculations, as it should due to diffeomorphism invariance. For convenience we further
define
QV =
(λV + 2)1/4
2
√
2
√
λV − 2
ΠQ˜V . (5.64)
The second order action for Q˜V reads
Sˇ(2) = 4π2L2
∫ ∞
0
dy
y30
√
fΠ2√
2− ym2y√y
[
(2− y)y(
λV − 2+ 4Π2
)
f
Q˜′V
2
+
Q˜2V
y20
]
. (5.65)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Since the above quadratic action is manifestly positive for any value of y0, there are no negative
modes in the vector-derived sector with V  2.
5.4.5. The vector V = 1 mode. The situation here is completely analogous to that of the
U(1)3 Maxwell theory case. This mode turns out to give a zero-mode when written in gauge-
invariant variables. So since the theory admits a symmetric hyperbolic formulation, it must be
the linearization of a pure-gauge mode and can be ignored.
5.4.6. The tensor modes with T  2. Finally, we come to the easiest sector, which is the
one defined by tensor-derived perturbations. The reason why this sector is easiest is twofold:
the scalar perturbations are zero in this sector, and the metric perturbation reduces to a single
gauge-invariant variable. Unlike in the U(1)3 case, the tensors to not excite vectors or scalar
derived perturbations. The metric perturbation takes the rather simple form
δ ds2T =
2
√
2y20√
1+ T
L2
(1− y)2HT (y)S
T
i j , (5.66)
where the factors multiplying HVT are only there for later convenience in the presentation of
the second order action. The second order action reads
Sˇ(2) = 4π2L2
∫ ∞
0
dy
y20
√
f
(1− y)2√2− y√y
[
(2− y)y
f
H′ 2T +
1
y20
(
λT + 2+ 4Π2
)
H2T
]
,
(5.67)
which is manifestly positive for all wormhole solutions.
6. Einstein-U(1)2 wormholes in 11-dimensional supergravity
Having explored wormholes in simple but ad hoc low-energy theories in sections 4 and 5,
we now wish to understand the behavior of at-first-sight similar wormholes in various UV-
complete theories. This will be explored in the next few sections, and will in particular raise
the important issue of possible brane instabilities.
We begin in the current section by describing a truncation of 11-dimensional supergravity
that leads to a four-dimensional action similar to that studied in section 4, but with only two
Maxwell fields instead of three. In later sections we will also consider an example in a mass-
deformed ABJM setup and a type IIB compactification that leads to a theory of scalar fields
in AdS3.
To begin our discussion, recall that bosonic fields of 11-dimensional supergravity are just a
metric (11)g and a four-form F(4) = dA(3). The Euclidean action reads
S = −
∫ (
(11)R111− 12F(4) ∧ 11F(4) −
i
6
F(4) ∧ F(4) ∧ A(3)
)
, (6.1)
with (11)R ≡ (11)RABGAB being the 11-dimensional Ricci scalar associated with (11)g and (11)RAB
its Ricci tensor. The equations derived from (6.1) are
(11)RAB −
1
2
(11)gAB
(11)R =
1
12
[
F(4)ACDEF(4) b
CDE − 1
8
(11)gAB F(4)CDEFF(4)
CDEF
]
,
(6.2a)
43
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
d  F(4) =
i
2
F(4) ∧ F(4). (6.2b)
Here upper case Latin indices are eleven-dimensional and 11 is the eleven-dimensional
Hodge dual operation.
6.1. Einstein-U(1)2-theory
We consider an ansatz where the eleven-dimensional fields take the form
ds211d ≡ (11)gAB dXA dXB = gab dxadxb
+
1
g2
{
dξ2 +
cos2 ξ
4
[
dθ21 + sin
2 θ1 dφ21 +
(
dψ1 + cos θ1 dφ1 − 2gA(1)
)2]
+
sin2 ξ
4
[
dθ22 + sin
2 θ2 dφ22 +
(
dψ2 + cos θ2 dφ2 − 2gA(2)
)2]}
, (6.3a)
with ξ ∈ (0, π/2), θi ∈ (0, π), φi ∈ (0, 2π) and ψi ∈ (0, 4π), and
F(4) = −6giVol4 + iF˜(4) (6.3b)
with
F˜(4) =
cos ξ
2g2
[
sin ξ dξ ∧ (dψ1 + cos θ1 dφ1 − 2gA(1))+ 12 cos ξ sin θ1 dθ1 ∧ dφ1
]
∧ F(1)
− sin ξ
2g2
[
cos ξ dξ ∧ (dψ2 + cos θ2 dφ1 − 2gA(2))− 12 sin ξ sin θ2 dθ2 ∧ dφ2
]
∧ F(2),
(6.3c)
where  is the Hodge dual with respect to the four-dimensional metric g, F(I) = dA(I) with
I = 1, 2 and Vol4 the volume form of g. Here, lower case Latin indices are four-dimensional.
We also restrict to configurations where
F(1) ∧ F(1) = F(2) ∧ F(2). (6.4)
Inserting the ansätze for the eleven-dimensional metric (11)g and four-form field F(4) into
the equations of motion (6.2) induces a set of equations for the four-dimensional metric g and
gauge fields F(I) which can be derived from the following somehow familiar four-dimensional
action
S = −
∫
M
d4 x
√
g
(
R+
6
L2
−
2∑
I=1
F(I)abF
(I) ab
)
− 2
∫
∂M
d3x
√
h K + SB, (6.5)
where the boundary terms are exactly as in equation (4.1) and 2g = L. This looks remark-
ably similar to the Einstein-U(1)3 theory, but with only two Maxwell fields. Any solution of
the equations of motion induced by (6.5) can be uplifted to a solution of eleven-dimensional
44
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
supergravity via equation (6.3). This truncation is a sub-truncation of a more general truncation
that (to our knowledge) first appeared in [62]. We will later consider in section 7 yet another
sub-truncation of [62].
We are interested in finding solutions whose boundary metric is a round S3. However, we
are going to relax the assumption that the SO(4) symmetry is preserved in the bulk. In fact, in
the bulk, we will only require our geometry to enjoyU(2) symmetry. The best way to visualise
how we are going to do this in a simple manner is to again introduce the left-invariant one
forms σˆi described in section 4 and to consider a line element of the form
ds2 =
dr2
f (r)
+
g(r)
4
[
h(r)σˆ23 + σˆ
2
1 + σˆ
2
2
]
, (6.6)
where the U(2) = U(1)× SU(2) symmetry is manifest (with the U(1) parametrising the angle
that rotates σˆ1 into σˆ2). The functions f , g and h are functions of r only. For the gauge fields
we take
A(1) =
L
2
Φ(r)σˆ1 and A
(2) =
L
2
Φ(r)σˆ2. (6.7)
We want to construct solutions where the dual operator to A(I) has a non-vanishing source.
This is obtained by searching for solutions for which
lim
r→+∞
Φ = Φ0. (6.8)
The objective of the sections below is to construct the phase diagram of the wormhole and
disconnected solutions as a function of Φ0.
6.2. The disconnected phase
The disconnected phase is easy to find analytically. Just as for the Einstein-U(1)3 theory, it
satisfies
F(I) = ±  F(I), (6.9)
with the lower sign yielding a singular solution. We thus take the upper sign.
The stress energy tensor is then identically zero, and g is just the usual metric on Euclidean
AdS with a boundary S3 for which
g(r) = r2, f (r) =
r2
L2
+ 1, h(r) = 1 and Φ(r) = Φ0
√
r2 + L2 − L√
r2 + L2 + L
.
(6.10)
We were also able to find solutions where h 
= 1, but it turns out they all lead to boundary
metrics that are not round spheres; i.e., the squashing does not disappear on the boundary. We
will comment further on this more general case later. It is a simple exercise to evaluate the
on-shell action for which we find
ΔSU(1)2 = 8π
2L2
(
1+ 2Φ20
)
. (6.11)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
6.3. The wormhole phase
Despite our best efforts we were not able to find an analytic solution for the wormhole phase.
We thus proceed numerically. Our ansatz takes the form
ds2 =
L2
(1− y)2
[
f dy2
y(2− y) +
y20
4
(
gσˆ23 + σˆ
2
1 + σˆ
2
2
)]
, (6.12a)
with f and g to be determined numerically and depending only on y. Just as for the Einstein-
U(1)3 theory, y0 measures the minimal size of the wormhole at the neck.
For the Maxwell fields we again take
A(1) =
L
2
Φ(y)σˆ1 and A
(2) =
L
2
Φ(y)σˆ2. (6.12b)
The equations of motion read
√
g
√
2− y√y√
f
[√
g
√
2− y√y√
f
Φ′
]′
− 4
y20
Φ = 0, (6.13a)
f g
nyyy20
+
8 f m2yΦ
2
nyyy40g
+
g′
gmy
− 1
nyyy20m
2
y
[(
4m2y + 3y
2
0
)
f − nyy
(
3y20 − 2m4yΦ′2
)]
= 0,
(6.13b)
3 f gm2y
ny
− 8 fΦ
2m4y
gy20ny
− f
(
4m2y + 3y
2
0
)
ny
− ymyy
2
0 f
′
f
− myyy
2
0
ny
+ (1+ 2y)y20 + 2m
4
yyΦ
′2 = 0
(6.13c)
with ny = 2− y and my = 1− y.
We now discuss the boundary conditions at y = 0, the wormhole Z2 plane of symmetry.
Demanding that f and g have a regular Taylor expansion around y = 0 gives the following set
of Dirichlet conditions at y = 0
Φ(0) =
√
g(0)y0
√
3y20 + 4− g(0)
2
√
2
,
Φ′(0) =
y0
√
6y20 + 8− 2g(0)√
g(0)
[
3y20 + 4− 2g(0)
] ,
f (0) =
y20
3y20 + 4− 2g(0)
,
g′(0)+
2g(0)
[
3y20 + 8− 5g(0)
]
3y20 + 4− 2g(0)
= 0.
One can show that Φ, f and g admit a regular Taylor series around y = 0, with all higher order
terms in the series being uniquely fixed by y0 and g(0).
We now turn out attention to the boundary conditions imposed at the conformal boundary
located at y = 1. There, we demand f (1) = g(1) = 1 since we are primarily interested in solu-
tions with a round S3 at the conformal boundary.Φ(1) ≡ Φ0, on the other hand, determines the
46
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
source. Expansion the fields with a regular Taylor expansion around y = 1 determines f , g and
Φ as a function of four unknowns which we take to be y0, Φ(1), Φ
′(1) and g′′′(1).
The procedure is now clear: we take a given value of y0 and g(0) and integrate outwards to
the conformal boundary and, in general, we find that g(1) 
= 1. This means for a given value of
y0, we need to adjust g(0) so that g(1) = 1. Once this is the case, we read off the correspond-
ing values of Φ0, Φ
′(1) and g′′′(1). We thus have a one-parameter family of solutions whose
boundary metric is a round three-sphere. This one-parameter family of solutions is, in turn,
labeled by y0.
Once we have the desired solution, we can determine its on-shell action numerically just
as we did when we studied the Einstein-U(1)3 theory. We call the Euclidean action of the
wormhole solution SW
U(1)2
.
6.4. M2 branes on wormhole backgrounds
Before discussing the phase diagram, we will pause for a moment and study probe branes on
wormhole backgrounds. The point of this study is that in the limit of weak coupling the action
of a probe brane describes the change in the action of the wormhole when a brane is inserted
into the solution11. As a result, negative probe-brane actions mean that than our wormhole is
not in fact the lowest-action solution. In this case one would in principle then like to study
solutions with such branes to find the true minimum, and to determine whether it remains a
connected wormhole or whether it becomes disconnected. However, this is beyond the scope
of the present work. At weak coupling such a true minimum can be achieved only by including
a large number of branes, which one might describe as a condensate. If on the other hand
all probe branes have positive action in our wormhole, this would support the idea that our
wormhole does indeed dominate the computation of the desired partition functions.
The appropriate M2 brane probe action is
SM2 = TM2
∫
MM2
d3σ
[√
det G˜− i εM2CM2
]
, (6.14)
with the metric on the world-volume of the M2 branes being given by
G˜μ˙ν˙ =
(11)gAB
dx A
dσμ˙
dx B
dσν˙
, (6.15)
and εM2 = ±1 for brane anti-brane configurations, respectively. Furthermore, the potential
term is given by
CM2 = 13!ε
μ˙ν˙ρ˙A(3) ABC
dx A
dσμ˙
dx B
dσν˙
dxC
dσρ˙
, (6.16)
with εμ˙ν˙ρ˙ being the totally anti-symmetric alternating symbol with ε1˙2˙3˙ = 1.
We are interested in branes that wrap the S3 so that we take σμ˙ = {ψ, θ, ϕˆ}with the standard
Euler angles given in (4.6). Applying this procedure to the our wormhole ansatz gives
SM2(y; εM2) = 2π2L3y30TM2
[ √
g(y)
(1− y)3 − 3εM2
∫ y
0
√
f (y˜)
√
g (y˜)
(1− y˜)4√y˜√2− y˜ dy˜
]
. (6.17)
11 This property, together with the idea that sources at the AdS boundaries should remain fixed, determines the detailed
form of the probe brane action and forbids the addition of arbitrary constants. With this understanding, the sign of the
probe-brane action becomes physically meaningful.
47
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
In deriving the above expression we made a choice in determining A(3) from F(4). This choice
was such that upon the change of variable
r = L y0
√
y
√
2− y
1− y (6.18)
the action SM2(r; εM2) satisfies
SM2(r;−1) = SM2(−r; 1), (6.19)
so that studying SM2(r; 1) covers both the case of brane and anti-brane probes. The question is
then whether SM2(r; 1) is positive definite for all values of r. If SM2(r; 1) < 0 for any range of
r, we would expect brane nucleation to take place and render our wormhole solution unstable.
6.5. Negative modes
Studying negative modes of this novel class of geometries turns out to be more complicated
than in the Einstein−U(1)3 case as even the backgroundmetric fails to enjoy spherical symme-
try. However, the current isometry group is now SU(2)× U(1) remains large enough to reduce
the study of the perturbations to ordinary differential equations. We have used this observa-
tion to analyze such perturbations by generalizing the techniques used for SO(4) symmetry
in the above sections. However, due to the extremely cumbersome nature of this procedure,
we present only a sketch of the analysis below. This sketch should suffice to allow dedicated
readers who wish to check and reproduce our results to do so.
We first introduce charged scalar harmonics on CP1 ≡ S2. Following [56] we define these
to be solutions of the following eigenvalue problem
DiDiYmκ + λmκYmκ = 0, (6.20)
with D = ∇ − imA
CP1 , m ∈ Z, ∇ being the standard connection on CP1 and ACP1 is the
Kähler one-form that relates to the standard Kähler two-form on CP1, J
CP1 , as J = dACP1/2.
Regularity then demands
λmκ = (+ 2)− m2 with  = 2κ+ |m| where κ = 0, 1, 2, . . . (6.21)
The construction of the metric and gauge field perturbations then conformswith those stud-
ied in [56]. Using numerics, we find that there are no negative modes for anym 
= 0 and κ > 0.
Form = κ = 0, however, for small wormholes we do find a negative mode on which we report
further below.
6.6. Results
Our numerical results are as follows. For each value of the boundary source Φ0 > Φ
min
0 ≈
4.0162(5) we find two wormhole solutions which we may again call small and large; see
left-hand side panel of figure 12. We find no field-theoretic negative modes anywhere on the
large wormhole branch, though the small wormhole branch has at least one negative mode
in the κ = m = 0 sector. Furthermore, the Euclidean action of the large wormhole branch
of solutions eventually becomes smaller than twice the corresponding action of the discon-
nected solution; see right-hand side panel of figure 12 where ΔSU(1)2 = 2S
D
U(1)2
− SW
U(1)2
—for
Φ0 > Φ
HP
0 ≈ 4.352(8).
Finally, we also studied the positivity of the probe M2 brane action (6.17). On the large
wormhole branch, and for large enough boundary sources Φ0 > Φ
M2
0 ≈ 4.097(2), we find a
48
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 12. Left panel: radius of the wormhole solutions as a function of the source
Φ0. Wormholes only exists for Φ0 > Φmin0 ≈ 4.0162(5). Right panel: difference in the
Euclidean action ΔSU(1)2 = 2S
D
U(1)2
− SW
U(1)2
as a function of Φ0. The wormhole solu-
tions have a lower action forΦ0 > ΦHP0 ≈ 4.352(8). The orange squares represent small
wormholes, the blue disks correspond to large wormholes where the Euclidean action
forM2 is positive and the green diamonds indicate large wormholes where the Euclidean
action for M2 is not positive definite. The red disk indicates the value of Φ0 in the large
wormhole branch above which the Euclidean action (6.17) for M2 branes is not positive
definite.
finite range of r (or, equivalently, values of y) for which the action from M2 branes wrapped
on the S3 becomes negative. As a result, all large wormholes which dominate over our discon-
nected solution turn out to be unstable to brane nucleation. We mark the nucleation threshold
atΦM20 with a red dot in the right panel of figure 12. We note, however, that there exists a range
Φ0 ∈ (Φmin0 ,ΦM20 ) in which the large wormholes seem to have no pathology, though in this
range they are subdominant with respect to our disconnected solution (these non pathological
wormholes are represented by the blue disks in figure 12).
Finally,we comment on a small extension of our result.We have also considered caseswhere
the metric at the boundary is not round, i.e. g(1) ≡ g1 
= 1. These constitute a two-parameter
family of wormhole solutions which we can parametrise with (Φ0, g1). It turns out that near
the conformal boundary, located at y = 1, one has
SM2(y; 1) =
π2(4− g1)√g1L3y0
1− y +O(1). (6.22)
It is thus clear that we need 0 < g1 < 4 in order for SM2(y; εM2) to be positive definite near
the conformal boundary. So we restricted our attention to 0 < g1 < 4. In this range we were
not able to find any value of g1 for which wormholes dominate over the disconnected solution,
have positive definite SM2(y; εM2), and have no negative modes. In fact, we find that for small
enough values of g1 even the small wormhole branch has non positive SM2(y; εM2). The value of
g1 corresponding to the largest ratioΦ
M2
0 /Φ
HP
0 is g1 = 4/3, corresponding to the maximisation
of the numerator in the divergent term of SM2(y; 1) near the conformal boundary.
49
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
7. Mass deformation of ABJM
Perhaps the simplest asymptotically-AdS Euclidean wormholes are the quotients of Euclidean
AdSd, which is of course also the hyperbolic plane Hd. This space admits a foliation by
hyperbolic planes Hd−1 of dimension (d − 1), so that the metric can be written
ds2 =
dr2
r2
L2
+ 1
+ (r2 + L2)ds2
Hd−1 , (7.1)
with ds2
Hd−1 the metric on the unit H
d−1. Here r takes values in (−∞,∞). Now, any compact
hyperbolic space of dimension d − 1 can be written as the quotient of Hd−1 with respect to
an appropriate discrete group Γ of Hd−1 isometries. From (7.1), we see that taking the corre-
sponding quotient of AdSd yields a wormhole with compact hyperbolic slices at each value of
r ∈ (−∞,∞) and with two separate boundaries at r = ±∞. In the obvious conformal frame
both boundaries are again compact hyperbolic manifolds.
This construction embeds easily in many UV-complete models. However, as described in
[10], in simple models it is associated with a dramatic brane-nucleation instability that occurs
even near the AdS boundary. This makes the entire theory unstable with such boundaries. The
dual field theory interpretation is that conformal field theories require conformal couplings to
curvature, and that such couplings naturally generate negative mass terms when the theory is
placed on a compact hyperbolic space.
As noted in [10], this also suggests that such instabilities can be cured by breaking con-
formal invariance and adding explicit new couplings to the would-be-CFT that give masses to
various scalars. The question is then what happens to the abovewormhole solutions under such
deformations. We investigate this issue below using a particular mass-deformation of an AdS4
compactification of 11-dimensional supergravity; i.e., from the dual gauge theory point of view
we study a deformation of the ABJM model [39]. Interestingly, at least in this case, we find
wormhole solutions only for small mass deformations μ, and in particular only at deforma-
tions μ < μmax where μmax is still too small to stabilize the theory. At such small deformations
we find two branches of wormhole solutions, but they coalesce μ = μmax. It thus appears that
wormholes do not exist in the stable members of this family of theories.
7.1. The mass-deformation model
The deformation of interest can be described using a sub-truncation of the truncation of 11-
dimensional supergravity detailed in [62]. Our 11-dimensional metric G takes the form
ds211d ≡ (11)gAB dx A dx B = Ξ1/3gab dxa dxb
+
Ξ1/3
g2
{
dξ2 +
cos2 ξ
4Z1
[
dθ21 + sin
2 θ1 dφ
2
1 + (dψ1 + cos θ1 dφ1)
2
]
+
sin2 ξ
4Z2
[
dθ22 + sin
2 θ2 dφ22 + (dψ2 + cos θ2 dφ2)
2
]}
, (7.2)
where
Z1 = eΦ cos2 ξ + sin2 ξ, (7.3a)
Z2 = (e−Φ + χ2eΦ)sin2 ξ + cos2 ξ, (7.3b)
Ξ = Z1Z2. (7.3c)
50
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
For F(4) we have
F(4) = −2igUVol4 + i sin ξ cos ξg (dΦ− χ e
2Φ  dχ) ∧ dξ − dA˜(3), (7.4)
where Vol4 is the volume form on g and  the Hodge operation with respect to g. Furthermore,
A˜(3) =
1
8 g2
χeΦ
[
cos4 ξ
Z1
sin θ1(dψ1 + cos θ1 dφ1) ∧ dθ1 ∧ dφ1
− sin
4 ξ
Z2
sin θ2(dψ2 + cos θ2 dφ2) ∧ dθ2 ∧ dφ2
]
(7.5)
and
U = eΦ cos2 ξ + (e−Φ + χ2eΦ)sin2 ξ + 2. (7.6)
Inserting the above expressions forG and F(4) into the eleven-dimensional equations of motion
(6.2) yields four-dimensional equations of motion for g, χ and Φ that can be derived from the
following four-dimensional action
S4d = −
∫
M
d4 x
√
g
(
R+
1
2
∇aΦ∇aΦ− e
2Φ
2
∇aχ∇aχ− V
)
− 2
∫
∂M
d3x
√
h K + SB, (7.7)
with
V = − 1
L2
(
eΦ + e−Φ + χ2eΦ + 4
)
, (7.8)
where 2g = L. In the above we will not need to specify SB. From the four-dimensional per-
spective, Φ and χ have masses μ2L2 = −2. To proceed we need to understand what boundary
conditionswe should choose for these fields. By comparing our ansatz forF(4) with the one pre-
sented in [63], we conclude that the sources associatedwith boundaryvalues of the supergravity
fieldsΦ andχ parametrise (possibly supersymmetric)mass deformations of ABJM. The super-
symmetric mass deformation corresponds to deformations for which χ ∼ μz/L, Φ = O(z2),
with μ being proportional to the mass deformation and z a Fefferman–Graham coordinate.
These are the boundary conditions that we will employ.
Rather remarkably, the action above admits yet another sub-truncation in which
χ =
√
1− e−2Φ. (7.9)
Perhaps more interestingly, when we performed our numerical studies, we did not impose the
relation above, and yet all our numerically determined solutions were consistent with the above
relation.
7.2. Wormholes
Our starting point is family of solutions (7.1) with d = 4. Below, we implicitly assume a quo-
tient by someΓ that makes the r = constant slices compact hyperbolic spaces. As noted above,
such solutions are unstable to the nucleation of branes, which the in present context are M2
branes. To understand the effect of the mass deformation on this instability we will study M2
51
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
branes on our deformed wormholes. Before doing this, we provide some detail regarding our
construction.
As a metric ansatz we take
ds2 =
L2
(1− y)2
[
f dy2
y(2− y) + y
2
0 ds
2
H3
]
, (7.10)
with f a function only of y ∈ (0, 1) and with y0 to be interpreted as the minimal size of the
wormhole neck. The neck is located at y = 0 and the conformal boundary is at y = 1. For the
scalars we take
χ = (1− y)q1 and Φ = (1− y)2q2. (7.11)
The procedure is now very similar to what we have seen before: finiteness of f at y = 0
locks y0 into a relationship with Φ(0) and χ(0). We then take these values at the origin y = 0
and integrate outwards. In general Φ = O[(1− y)] so we adjust Φ(0) so that near the confor-
mal boundary we find Φ = O[(1− y)2]. What remains is a one-parameter family of solutions
parameterized by either y0 or μ.
7.3. M2 probe branes
We now need to understand what value of μ is required to remove the UV brane-nucleation
instability by making the action for probe M2 branes positive near the AdS boundary. We
use the action (6.17) together with the 11-dimensional ansätze (7.2) and (7.4). We begin by
wrapping our M2 branes on (the relevant quotient of) H3, and obtain an action as a function
of y. Near the conformal boundary we can use the near boundary behavior of our fields to
determine SM2 near y = 1. This turns out to be
SM2 =
y0L3 VolH3
8(1− y)
[
μ2 − 12+ 2 cos(2ξ) (μ2 − y20Φ′′(1))]+O(1), (7.12)
where VolH3 is the volume of the relevant quotient of H
3. The condition (7.9) automatically
ensures that the term proportional to cos(2ξ) vanishes (we once more note that we find this
condition to be true numerically). We thus see that we must have μ2 > 12 to stabilize the
theory in the UV, and in particular to have a hope of SM2 being positive definite.
7.4. Results
The main result of this section is presented in figure 13, where we plot the radius of the
wormhole y0 as a function of μ. Rather interestingly, we find no wormhole solution for
μ > μmax ≈ 1.66(2), which is smaller than
√
12 ≈ 3.464, so that this new class of mass
deformed wormholes is still unstable to nucleating M2 branes. The results of this section are
thus similar to those found in [27] for mass deformations ofN = 4 SYM.
8. Wormholes from type IIB theory
We can also find a truncation of a UV-complete scenario that is closely related to our Einstein-
scalar model of section 5, though this truncationwill live in AdS3 instead of AdS4. As described
below, this model is a compactification of type IIB supergravity. After discussing the ansätze
for the fields in section 8.1, we analyze the model as usual in the remaining subsections.
52
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 13. Nowormhole solutions seem to exist for μ > μmax ≈ 1.66(2) <
√
12, which
in particular implies that no stable wormhole solution exist.
8.1. A simple consistent truncation
We consider type IIB supergravity with only the ten dimensional metric (10)g, Ramond–
Ramond three-form F(3) ≡ dA(2) and dilaton φ. The corresponding equations of motion are
(10)RAB =
1
2
(10)∇Aφ(10)∇Bφ+
eφ
4
[
F(3) ACDF(3) B
CD −
(10)gAB
12
F(3) CDEF(3)
CDE
]
,
(8.1a)
d
(
eφ10F(3)
)
= 0, (8.1b)
φ− e
φ
12
F(3) ABCF(3)
ABC = 0, (8.1c)
where 10 is the Hodge operation associated with the ten-dimensional metric (10)g,
(10)∇ its
associated metric-compatible connection and upper case Latin indices are ten-dimensional.
Consider the following ten-dimensional field configuration
ds2 = (gab dx
a dxb + L2 dΩ23)e
φ4
2 + e−
φ4
2
[
(e
√
2φ1 dz21 + e
−√2φ1 dz22)e
φ3
+ (e
√
2φ2 dz23 + e
−√2φ2 dz24)e
−φ3
]
(8.2a)
F(3) =
2i
L
Vol3 + 2L2 d3Ω3 (8.2b)
and
φ = φ4, (8.2c)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
where d3Ω3 is the volume form on a round 3-sphere, Vol3 is the volume form of the three-
dimensional metric g, φi, with i = 1, 2, 3, 4, depend only on the three-dimensional coordinates
xa and the coordinates {z1, z2, z3, z4} parametrise a four-torus. Here, lower case Latin indices
are three-dimensional. Inserting the ansätze (8.1) into the ten-dimensional equations of motion
(8.1) yields a set of three-dimensional equations for g, and φi which can be derived from the
following three-dimensional action
S3d = −
∫
M
d3x
√
g
[
R+
1
L2
−
4∑
i=1
∇aφi∇aφi
]
. (8.3)
This action will have to be supplemented by appropriate boundary terms. As expected, the
three dimensional metric g is asymptotically AdS3. The scalars φi appear as massless AdS3
scalars that are minimally coupled to gravity.
We wish to turn on a source for all the φi, and we should thus consider adding boundary
terms to (8.3) appropriate to such a choice and which render the on-shell action finite on solu-
tions to correspondingequations ofmotion.We now introduceFefferman–Graham coordinates
for g, where the conformal boundary is located at z = 0. Since φi is massless, it will have in
general a log z divergence close to the conformal boundary. In fact, the generic behavior of φi
close to the conformal boundary takes the form
φi = Vi(xμ)+ z2Zi(xμ)+ z2 log zZ˜i(xμ)+ · · · , (8.4)
where μ runs over the boundary directions, and Z˜i is a function of Vi(xμ) only. For this reason,
the counterterms to be added to (8.3)will explicitly depend on aUVcut off z = ε. The boundary
terms that lead to a well defined variational problem for the metric and scalar field and render
the on-shell action finite read
Sc(ε) = −2
∫
∂Mε
d2x
√
h K +
2
L
∫
∂Mε
d2x
√
h
− L log ε
L
4∑
i=1
∫
∂Mε
d2x
√
h∇˜μφi∇˜μφi − L log εL
∫
∂Mε
d2x
√
hR˜, (8.5)
where h the inducedmetric on a surface of constant z = ε. We denote this surface by ∂Mε, and
∇˜ is the natural metric-compatible connection on (∂Mε, h). Furthermore, R˜ the Ricci scalar
on ∂Mε and K the trace of the extrinsic curvature associated to an outward-pointing normal
to ∂Mε. The first term in (8.5) is the usual Gibbons–Hawking–York term, and the remaining
are boundary counterterms. The total on-shell action is then
Son−shell = lim
ε→0+
S˜(ε)+ Sc(ε), (8.6)
where S˜(ε) is obtained from (3)S in (8.3) by replacingM withMε, which is obtained fromM
by chopping off from spacetime the region z > ε.
In the remainder of this section we will set φ4 = 0, which in particular restricts us to trivial
dilaton fields. We stress however, that appendixC describes wormhole solutions with a toroidal
boundary metric T2 for which φ4 
= 0.
8.2. Symmetry
The solutions we seek to construct enjoy spherically symmetric metrics, but the scalars φi will
break said symmetry in a special way. In particular, we introduce standard polar and azimuthal
54
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
coordinates (θ,ϕ) on the S2, so that the standard metric on the S2 reads
dΩ22 = dθ
2 + sin2 θ dϕ2. (8.7)
We then take the scalars to satisfy
φ1 = ψ(r) sin θ cos ϕ, φ2 = ψ(r) sin θ sin ϕ and φ3 = ψ(r) cos θ.
(8.8)
These forms are chosen so that
3∑
i=1
φ2i = ψ(r)
2. (8.9)
For the metric we take
ds2 =
dr2
f (r)
+ g(r)dΩ22. (8.10)
Wormhole solutions will have f (r), g(r) > 0 throughout spacetime, whereas the discon-
nected solutions will have f (r) = 1+O(r2), with g(r) = O(r2) near r = 0.
8.3. D1 branes
We again wish to address potential brane nucleation instabilities. Since we are in type IIB and
the only non-zero Ramond–Ramond field is the three-form F(3), the relevant branes are D1’s
and D5’s. We shall present here results for the D1’s, but the invariance of the background under
Hodge-duality of the Ramond–Ramond three-form field strength and the trivial dilaton imply
that equivalent results can be obtained for D5’s wrapped on the four-torus (or for related D1D5
bound states).
For a generic spacetime (M, g,A(2)), the Euclidean action of a probeD1 with world-volume
coordinates σμ˙ (for μ˙ = 1, 2) takes the simple form
SD1±E =
∫
M2
d2 σ
[√
det
(
gAB
∂x A
∂σμ˙
∂x B
∂σν˙
)
± i
2
εμ˙ν˙
∂x A
∂σμ˙
∂x B
∂σν˙
A(2) AB
]
, (8.11)
where the lower and upper signs stand for D1 and D1, respectively. We will choose our
brane world-volume coordinates to wrap the S2, so that σ1˙ = θ, σ2˙ = ϕ. We also introduce
coordinates on the S3 for which the metric is
dΩ23 = dθ˜
2 + sin2 θ˜
(
dθˆ2 + sin2 θˆ dϕ˜2
)
, (8.12)
with θ˜, θˆ ∈ (0, π) and ϕ˜ ∈ (0, 2π).
Within our symmetry class, we can write A(2) locally as
A(2) =
2i
L
λ(r) sin θ dθ ∧ dϕ+ L2λ˜(θ˜) sin θˆ dθˆ ∧ dϕ˜, (8.13)
with
dλ
dr
=
g(r)√
f (r)
, and
dλ˜
dθ˜
= sin2 θ˜. (8.14)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
One can thus write SD1E as
SD1± E = 4π
[
g(r)∓ 2
L
λ(r)
]
. (8.15)
To simplify our analysis, we take as a boundary condition λ(0) = 0 for both the wormholes
and disconnected solutions. Using these boundary conditions, it is easy to check that λ is an
odd function of r. Thus, if we check that SD1E > 0 for the upper sign and for all values of r, we
automatically guarantee positivity for the lower sign as well.
Even without actually solving the equations of motion for all values of r, we can obtain
useful information by studying the asymptotic behavior of solutions. We first map everything
to Fefferman–Graham coordinates, defined via
dr
dz
= −
√
f (r(z))
L
z
(8.16)
with rz = L2 as z→ 0. It is also useful to define
g =
L2
z2
G, (8.17)
in terms of which the equation for λ becomes:
dλ
dz
= −L
3
z3
G. (8.18)
Solving the resulting equations of motion asymptotically yields
G = 1+
1
2
(
V2 − 1) z2 +O (z4) , (8.19a)
ψ = V + (λ(2) + V log z)z2 +O (z4) , (8.19b)
where λ(2) is a constant. Integrating (8.18) then gives
λ =
L3
2z2
− L
3
2
(V2 − 1) log z+O(1), (8.20)
which in turn yields
SD1+ E = L
2(V2 − 1) log z+O(1). (8.21)
This result suggests that any solution with V > 1 will be unstable to spontaneously nucle-
ating a D1 brane. The question is then whether we can find any wormhole solution with
V < 1. If so, we must then further check to see whether the probe brane action on the resulting
background is positive definite for all z (and not just near z = 0).
8.4. The disconnected phase
We introduce radial coordinates, for which g in (8.10) is given by
g(r) = r2. (8.22)
To work with compact coordinates, we also introduce a new coordinate y ∈ (0, 1) so that
r = L
y
√
2− y2
1− y2 , (8.23)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
with y = 1 being the location of the conformal boundary and y = 0 the center where the S2
shrinks to zero size. Regularity at r = y = 0 demands that
ψ(r) = O(r) = O(y). (8.24)
To sum up, we take the ansatz
ds2 =
L2
(1− y2)2
[
4 dy2
f˜ (y) (2− y2) + y
2(2− y2)dΩ22
]
and ψ = y
√
2− y2q(y),
(8.25)
where have translated the f (r) in (8.10) into a function f˜ (y) = f (r(y)). The Einstein and
Klein–Gordon equations yield
f˜ (y) =
8− y2 (2− y2) (1− y2)2[2 (1− y2) q(y)+ y (2− y2) q′(y)]2
8
[
1− y2 (2− y2) (1− y2)2 q(y)2] ,
(8.26a)
1− y2
y
√
f˜ (y)
⎡
⎣ y2(2− y2)3/2√
f˜ (y)
(
1− y2)
(
y
√
2− y2q(y)
)′⎤⎦
′
− 8q(y) = 0. (8.26b)
Note that equation (8.26b) is a second order differential equation for q, since f˜ is given in terms
of q and q′ in equation (8.26a). As boundary conditions, we take q′(0) = 0 (which follows from
regularity at y = 0) and q(1) = V .
8.5. The wormhole phase
For the wormhole phase, we take g in (8.10) to have the form
g(r) = r2 + r20 (8.27)
with r0 denoting the wormhole radius. As we shall see, it will correspond to the minimum size
of the S2. We change coordinates to
r = r0
√
y
√
2− y
1− y , (8.28)
where theZ2 symmetry plane of the wormhole solution is identifiedwith y = 0, and the confor-
mal boundary is located at y = 1. It is also convenient to define r0 ≡ y0L. The relevant ansatz
now reduces to
ds2 =
L2
(1− y)2
[
fˆ (y) dy2
(2− y)y + y
2
0 dΩ
2
2
]
, and ψ = q(y), (8.29)
where, once again, we rewrite f in (8.10) in terms of fˆ (y).
The Einstein equation and Klein Gordon equations now yield
fˆ (y) =
y (2− y) y20
[
2− (1− y)2q′(y)2]
2
[
y20 − (1− y)2
(
q(y)2 − 1)] , (8.30a)
57
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 14. Left panel: radius of the wormhole solutions as a function of the source V .
Wormholes only exists for V > Vmin ≈ 3.14. Right panel: difference in the Euclidean
action ΔS3d = 2SD3d − SW3d as a function of V. The wormhole solutions have a lower
action for V > VHP ≈ 3.7373(8). Unlike the higher-dimensional examples, we find no
evidence for a small wormhole branch.
√
y
√
2− y√
fˆ (y)
⎡
⎣ √y√2− y
(1− y)
√
fˆ (y)
q′(y)
⎤
⎦
′
− 2q(y)
y20(1− y)
= 0. (8.30b)
Reflection symmetry around y = 0, and smoothness of the corresponding solution imply
q(0) =
√
1+ y20, and q
′(0) = 2
√
1+ y20
y20
. (8.31)
At the conformal boundary, we wish to set q(1) = V . The strategy is now simple enough:
we take equation (8.30b) and integrate it all the way to y = 1, where we read off V for any
value of y0 we choose. We found it convenient to perform this integration using an implicit
fourth order Runge–Kutta method.
8.6. Results
There are a couple of surprising results in this setup. First, we only find one branch of wormhole
solutions, which we coin as large since it extends to arbitrarily large values of the source V .
Nevertheless wormholes only seem to exist for V  Vmin ≈ 3.14. These results can be seen
on the left-hand side panel of figure 14 where we plot y0 as a function of V . The fact that
V > 1 in order for the wormhole solution to exists immediately reveals that the wormholes we
found suffer from nucleation instabilities associated with D1 and D¯1 branes. Finally, on the
right-hand side panel of figure 14 we plot the different in on-shell action
ΔS3d = 2SD3d − SW3d, (8.32)
where SD3d and S
W
3d are the three-dimensional on-shell actions for the disconnected and worm-
hole solutions (respectively). We can see that the wormhole solution dominates over the
disconnected solution for V  VHP ≈ 3.7373(8).
58
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
We also studied the field theoretical stability of the wormhole solutions. The analysis is
completely analogous to section 5.4, except it is easier because there are no tensor harmonics
on S2 and vectors harmonics can be obtained from scalar harmonics via the Hodge operation
on gradients of the scalar harmonics. We found no negative modes, irrespectively of the value
of V .
9. Discussion
Our work above explored the construction and stability of asymptotically anti-de Sitter
Euclidean wormholes in a variety of models. Indeed, we have studied many more models
than were described above, but for brevity we limited our presentation to a few representa-
tive cases with spherical (or squashed sphere) boundaries. A few low energymodels with torus
boundaries are discussed in appendix C, and a table showing a longer list of 22 string/M-theory
compactifications and 14 ad hoc low energymodels and associated results obtained in a variety
of dimensions is presented in appendix D. While not all issues were analyzed for all models in
the table, we hope it will nevertheless be of use in guiding future studies. Perhaps notably, our
list does not include the model described in section 5 of [10] where the disconnected solution
remains to be found in order to determine if the wormhole described there will dominate.
In simple ad hoc low-energymodels, it was straightforward to find two-boundary Euclidean
wormholes that dominate over disconnected solutions and which are stable (lacking negative
modes) in the usual sense of Euclidean quantum gravity. Similar results were found previously
in the context of JT gravity coupled to matter [17] and in studies of constrained wormholes in
pure gravity [37]. In particular, resulting phase diagram was a direct analogue of the Hawk-
ing–Page phase transition for AdS–Schwarzschild black holes in which, for boundary sources
above some threshold, we find both a ‘large’ and a ‘small’ branch of wormhole solutions with
the latter being stable and dominating over the disconnected solution for large enough sources.
We also studied two-boundary Euclidean wormholes in a variety of string and M-theory
compactifications. At first glance the solutions are generally similar to those in the ad hoc
models, and we find several contexts where large wormholes dominate over the disconnected
solutions we find and where the large wormholes are stable with respect to field-theoretic per-
turbations. However, wormholes in these UV-complete settings that are large enough to dom-
inate over our disconnected solution always suffer from brane-nucleation instabilities (even
when sources that one might hope would stabilize such instabilities are tuned to large values).
This implies the existence of additional solutions with lower action. It is natural to expect that
the lowest-action such solutions are again disconnected, but this remains to be studied in detail.
Including finite back-reaction from such branes would be a natural next step in understanding
wormhole solutions in string theory.
The overall picture of UV-complete models is thus rather similar to that obtained by Mal-
dacena and Maoz [10] with the following exceptions. First, we have performed a thorough
analysis of potential field-theoretic negative modes and shown our large wormholes to be
free of such pathologies. Second, we have identified subdominant wormholes that are free
of both field-theoretic and brane-nucleation instabilities. In particular, this was the case for the
U(1)2 largewormholeswith round boundaries and 4.352(8) ≈ ΦM20 > Φ0 > Φmin0 ≈ 4.0162(5)
described section 6.6, though in other models subdominant wormholes can suffer from brane
nucleation instabilities as well (see e.g. section 8 and the discussion of general squashed
boundaries in section 6.6). Third, section 7 investigated finite values of sources that provide
mass-deformations of the form that were suggested in [10] to stabilize theories against brane-
nucleation.As predicted by [10], this does in fact stabilize the theory in the sense that it removes
the UV brane-nucleation instability near the asymptotically-AdS boundaries. One thus expects
59
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
that the disconnected solution is fully stabilized at such values of the deformation parameter.
However, we were able to find wormhole solutions only when the deformation is small enough
that the UV remains unstable. This is perhaps the strongest evidence yet that, at least with-
out taking parameters to exponentially large values (see below), Euclidean wormholes will not
dominate partition functions in UV-complete theories.
The interesting question is of course what such results imply for the AdS/CFT factorization
problemdescribed in the introduction.To begin this discussion, we note that a brane-nucleation
instability is really the statement that adding a brane to the given solution will lower the action.
Since branes are discrete, if these are the only ‘instabilities’ this means that sufficiently small
fluctuations around the solution must in fact increase the action. So in a technical sense the
wormhole saddles we found in the string-compactified models are in fact stable. The point is
simply that the wormholes constructed thus far will be sub-dominant saddles and will not
control leading-order effects.
This result is natural even if one believes that bulk AdS gravity in UV-complete theories
should be dual to an ensemble of quantum theories. Had we found a case where a simple
semi-classical wormhole dominates the computation of a partition function, this would have
indicated that ensemble-fluctuations of that partition function are large, or at least that they are
not particularly small when compared with its ensemble expectation-values12. In other words,
it would have implied that an ensemble dual to bulk string theory is not sharply peaked in the
semi-classical limit. On the other hand, evidence to date suggests that ensembles associated
with quantum gravity are generally peaked very sharply indeed. This is the case whether one
looks at the ensembles associated with low-dimensional gravitational wormholes [12–15, 26,
64] or at fluctuations in ensembles [65] of states associated with black hole interiors13. Indeed,
the fluctuations associated with the double cone solution [13] are visible only when one probes
the fine structure of the associated ensemble by studying exponentially large times. One might
similarly expect that more standard partition functions become dominated by wormholes only
at exponentially large sources, which is a regime that we have certainly not probed (and which
may involve additional UV physics due large field values and strong gradients)! It is interesting
that UV-completemodels appear to generally achieve this expectationwhile ad hoc low-energy
models often admit exceptions.
What then are the implications of our (sub-dominant) wormhole saddles in the string-
compactified models? We emphasize that they are in fact stable with respect to sufficiently
small perturbations (within the truncations studied), so that they cannot be immediately
dismissed. This is in particular a technical advance beyond the analysis performed in [10].
While in general the semi-classical approximation leads to a sum over all (stable) Euclidean
saddles, in many contexts it would be dangerous to draw conclusions about physics based on
sub-dominant saddles. This is simply because the effects of sub-dominant saddles are small,
and so inmany cases can be easily dwarfed by even small corrections to the physics of dominant
saddles.
However, the present context appears to be rather different. In studying quantities like
δZ2 := 〈Z2〉 − 〈Z〉2 the contribution of any disconnected saddle will vanish identically, and
without error. So only contributions from connected saddles remain. Unless such contribu-
tions fully cancel among themselves, δZ2 will be non-zero. Though we are certainly not able
to analyze the possibility of a conspiracy that might enforce such cancellations, even in the
12We thank Henry Maxfield for discussions on this point.
13 And indeed, these may be closely related [26].
60
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
string compactifications we studied the most naive interpretation of our sub-dominant worm-
hole saddles remains that they will make δZ2 non-zero and require an ensemble of dual field
theories. This is of course also the picture implied by the dominance that of the double-cone
wormholes of [13] that appears to hold in late time computations of the spectral form factor in
generic models14. The factorization problem of AdS/CFT thus remains far from resolved and
will surely be the object of much future investigation.
Acknowledgments
We thank Xi Dong, Jerome Gauntlett, Juan Maldacena, Henry Maxfield, Douglas Stanford
and Edward Witten for useful conversations. We also thank the Kavli Institute for Theoretical
Physics for its hospitality during a portion of this work. As a result, this research was sup-
ported in part by the National Science Foundation under Grant No. NSF PHY-1748958. DM
was primarily supported by NSF Grant PHY-1801805 and funds from the University of Cali-
fornia. JES is supported in part by Grant ST/T000694/1. JES was also supported by a J Robert
Oppenheimer Visiting Professorship at the Institute for Advanced Studies in Princeton.
Data availability statement
No new data were created or analysed in this study.
Appendix A. Symmetric matrices for the Einstein-U(1)3 wormholes
A.1. The symmetric matrix V for the scalars
The symmetric matrix V is given by
VIJ = (K−1)IK(K−1)JMVKM. (A.1)
We then have
V22 =
(S + 1)
64π2g2(λS − 3)λS
[
g2 (λS − 2)λS + 24Φ2
(
gL2λS + 4r20L
2 + 4r40
)]
.
(A.2)
The remaining two components take a more complicated form.
For V11 we find
V11 =
g (S + 1)
32π2L2r3 f (λS − 3)
(
gλS + 24L2Φ
2
)2
[
3∑
i=0
Φ2i p(i)11(r)+ΦΦ
′
2∑
i=0
Φ2im(i)11(r)
]
,
(A.3a)
14 Though it would be interesting to further investigate higher-dimensional double cones in detail.
61
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
where
p(3)11 (r) = −
192L8r (λS − 4) (λS − 3)
g
, (A.3b)
p(2)11 (r) = −
8L4r
g
[
6g2(λS − 4)(λS − 3)+ gL2(λ3S − 14λ2S + 72)
− 6gr2(λS − 4)λS + 3r2(r2 − L2)(λS − 4)λS
]
, (A.3c)
p(1)11 (r) = rλS
{
L2
[
2gr2λS (2λS − 13)− 2g2
(
λ2S − 14λS + 42
)
+ r4
(−2λ2S + λS + 12)]
+ L4
[
g
(
λ2S − 2λS − 60
)
+ r2
(
2λ2S − λS − 12
)]
+ 24gr40 (λS − 1)
}
, (A.3d)
p(0)11 (r) = −
rλ2S
8L2
{
L4
(
λ2S + 3λS − 18
)− g2 [12L2r20 (λS + 1)+ 12r40 (λS + 1)]
− g3L2 (λ2S + 3λS − 18)+ gL2r20 (L2 + r20) (λ2S − 3λS − 18)
+ 4r40
(
L2 + r20
)2
(λS + 1)
}
, (A.3e)
m(2)11 (r) = −384L6
(
2g+ L2
)
(λS − 3) , (A.3f)
m(1)11 (r) = −8L2λS
{
3g2L2 (λS − 3)+ g
[
12L2r20 − (λS − 6)L4 + 12r40
]
+ 5L2r20
(
L2 + r20
)
(λS − 3)
}
, (A.3g)
and
m(0)11 (r) = λ
2
S
[
3g3L2 (λS + 2)+ 3g
2L4λS + 3g
4 − gL2r20
(
L2 + r20
)
(3λS − 4)
+ r40
(
L2 + r20
)2]
. (A.3h)
While for V12 we have
V12 =
(S + 1)
32π2L2r2 f (λS − 3)
(
gλS + 24L2Φ
2
)2
[
3∑
i=0
Φ2i p(i)12(r)+ΦΦ
′
2∑
i=0
Φ2im(i)12(r)
]
,
(A.4a)
where
p(3)12 (r) = 2304L
6
(
2g+ L2
)
, (A.4b)
p(2)12 (r) = −96L2
{
g2L2(λS + 6)+ g
[
12L2r20 + L
4(λS + 6)+ 12r
4
0
]− L2r20(L2 + r20)λS} ,
(A.4c)62
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
p(1)12 (r) = −4λS
{
15g3L2 − g2 [6r20(L2 + r20)+ L4(λS − 15)]
+ gL2r20(L
2 + r20)(2λS − 21)+ 6r40(L2 + r20)2
}
, (A.4d)
p(0)12 (r) =
gλ2S
2L2
{
g2
[
6L2r20 + L
4(λS − 3)+ 6r40
]
+ g3L2(λS − 3)
− gL2r20(L2 + r20)(λS − 7)− 2r40(L2 + r20)2
}
, (A.4e)
m(2)12 (r) = −1152 f L8r, (A.4f)
m(1)12 (r) = 144 f
2L6r3λS, (A.4g)
and
m(0)12 (r) = 2grλ
2
S
[
g2(7L2 + 3r20)+ 3g
3 + g(4L2 − 3r20)(L2 + r20)− 3r20(L2 + r20)2
]
.
(A.4h)
A.2. The symmetric matrix V for the tensors
The symmetric matrix V is given by
VIJ = (K−1)IK(K−1)JMVKM. (A.5)
We then have
V11 =
64π2
25g2L2
{
25gL2(m− 1)m+ [6m(3m+ 4)− 17]r20
(
L2 + r20
)
− 24L4(m− 1)(3m+ 7)Φ2} , (A.6a)
V12 = − 32
√
2π2(m− 1)
125g2L2(6m− 1)
{
25gL2
[
8(1− 6m)Φ+ 17m(m− 1)2]
+ 17(m− 1) [8L4 (9m2 − 3m+ 19)Φ2 + (6m− 18m2 − 13)r20 (L2 + r20)]}
(A.6b)
V22 =
32π2(m− 1)2
625L4(g− 6gm)2
{
850 f gL4(1− 6m)2rΦ′ + 1250g2L2(1− 6m)2
+ 25g
[
36L2(1− 6m)2r20 − 144(1− 6m)2Φ2L4
− 340(12m2 − 8m+ 1)ΦL4 + 289m(m− 1)3L4 + 36(1− 6m)2r40
]
+ 289L2(m− 1)2 [(18m2 − 36m− 7)r20 (L2 + r20)− 8L4(3m− 8)(3m+ 2)Φ2]} .
(A.6c)
63
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Appendix B. Symmetric matrices for the scalars
The symmetric matrix V is given by
VIJ = (K−1)IK(K−1)JMVKM. (B.1)
We then have
VIJ =
1
fΠ2y40nym
2Δ
y
(
myΠ
′2 + λS − 3
)
dIJ
9∑
i=0
Π′iV(i)IJ . (B.2)
with V(9)11 = V
(9)
22 = 0, ny = y(2− y), my = 1− y and
d11 =
(
myΠ
′2 + λS − 3
)2
, (B.3)
d12 =
(
myΠ
′2 + λS − 3
) [
Π2
(
myΠ
′2 − 3)− myΠ′2 − λS + 3] , (B.4)
d22 = (λS − 3)
[
Π2
(
myΠ
′2 − 3)− myΠ′2 − λS + 3] . (B.5)
Furthermore,
V
(8)
11 = y
4
0
[
ΔΠ4 − (Δ− 1)Π2 + 1)]myn2y , (B.6)
V
(7)
11 = 2Π
3y40myn
2
y (Δ+ λS) , (B.7)
V
(6)
11 = Π
6
(
2 f y20m
2
ynyλS −Δ f y20m2yny − 4 f y20m2yny
)
+Π4
(
2Δ f y20m
2
yny − 3 f y20m2ynyλS + 8 f y20m2yny + 4Δy40myn2yλS
− y40myn2yλ2S + 3y40myn2yλS +Δ2y40myn2y − 12Δy40myn2y
)
+Π2
(
f y20m
2
ynyλS −Δ f y20m2yny − 3 f y40myny − 7 f y20m2yny
− 3Δy40myn2yλS + 4y40myn2yλS −Δ2y40myn2y + 12Δy40myn2y
− 6y40myn2y
)
+3 f y40myny+3 f y
2
0m
2
yny+3y
4
0myn
2
yλS−12y40myn2y , (B.8)
V
(5)
11 = Π
3
(
4 f y20m
2
ynyλS + 6Δy
4
0myn
2
yλS − 6y40myn2yλS − 18Δy40myn2y
)
− 2 fΠ5y20m2ynyλS, (B.9)
V
(4)
11 = Π
4
(
7Δ f y20m
2
ynyλS − 2 f 2m2yλS − 6 f y20m2ynyλ2S + 47 f y20m2ynyλS
− 18Δ f y20m2yny − 72 f y20m2yny + 2Δ2y40myn2yλS + 3Δy40myn2yλ2S
− 30Δy40myn2yλS + 6y40myn2yλ2S − 18y40myn2yλS − 9Δ2y40myn2y
+ 54Δy40myn
2
y
)
+Π6
(
4 f 2m2yλS − 4Δ f y20m2ynyλS + 4 f y20m2ynyλ2S
− 26 f y20m2ynyλS + 9Δ f y20m2yny + 36 f y20m2yny
)− 2 f 2Π8m2yλS
+Π2
(
3 f y20m
2
ynyλ
2
S − 3Δ f y20m2ynyλS − 21 f y20m2ynyλS
64
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
+ 9Δ f y20m
2
yny + 27 f y
4
0myny + 63 f y
2
0m
2
yny − 3Δ2y40myn2yλS
− 3Δy40myn2yλ2S + 27Δy40myn2yλS + 2y40myn2yλ2S − 15y40myn2yλS
+ 9Δ2y40myn
2
y − 54Δy40myn2y
)
+ 9 f y40mynyλS
+ 9 f y20m
2
ynyλS − 27 f y40myny − 27 f y20m2yny
+ 3y40myn
2
yλ
2
S − 27y40myn2yλS + 54y40myn2y , (B.10)
V
(3)
11 = Π
3
(
6Δ f y2y20mynyλS − 6 f 2y20m2yλS − 6 f 2m3yλS + 6 f y2y20mynyλS
+ 6Δ f y40mynyλS + 6Δ f y
2
0mynyλS − 12Δ f yy20mynyλS − 6 f y40mynyλ2S
+ 18 f y40mynyλS + 6 f y
2
0mynyλS − 12 f yy20mynyλS + 4Δy40myn2yλ2S
− 36Δy40myn2yλS + 6y40myn2yλ2S − 18y40myn2yλS + 54Δy40myn2y
)
+Π5(6 f 2m3yλS + 6 f
2y20m
2
yλS), (B.11)
V
(2)
11 = (9 f
2m2yλS − 3 f 2m2yλ2S)Π8 +
(
6 f 2λ2Sm
2
y − 108 f nyy20m2y − 27 fΔnyy20m2y
− 16 f nyy20λ2Sm2y − 3 fΔnyy20λ2Sm2y − 18 f 2λSm2y + 84 f nyy20λSm2y
+ 18 fΔnyy20λSm
2
y
)
Π6 +
(
27Δ2myn2yy
4
0 − 108Δmyn2yy40 +Δ2myn2yλ2Sy40
− 12Δmyn2yλ2Sy40 − 9myn2yλ2Sy40 − 12Δ2myn2yλSy40 + 72Δmyn2yλSy40
+ 27myn2yλSy
4
0 − f m2ynyλ3Sy20 + 40 f m2ynyλ2Sy20 + 6 fΔm2ynyλ2Sy20
+ 216 f m2ynyy
2
0 + 54 fΔm
2
ynyy
2
0 − 183 f m2ynyλSy20 − 36 fΔm2ynyλSy20 − 3 f 2m2yλ2S
+ 9 f 2m2yλS
)
Π4 +
(
27Δ2myn2yy
4
0 −Δmyn2yλ3Sy40 + 108Δmyn2yy40 + 54myn2yy40
− 3Δ2myn2yλ2Sy40 + 18Δmyn2yλ2Sy40 − 3 f mynyλ2Sy40 − 81 f mynyy40
+ 18Δ2myn2yλSy
4
0 − 81Δmyn2yλSy40 − 18myn2yλSy40 + 36 f mynyλSy40
+ 3 f m2ynyλ
3
Sy
2
0 − 33 f m2ynyλ2Sy20 − 3 fΔm2ynyλ2Sy20 − 189 f m2ynyy20
− 27 fΔm2ynyy20 + 135 f m2ynyλSy20 + 18 fΔm2ynyλSy20
)
Π2 − 108myn2yy40
+ 81 f mynyy40 + myn
2
yy
4
0λ
3
S + 81 f m
2
ynyy
2
0 − 18myn2yy40λ2S + 9 f mynyy40λ2S
+ 9 f m2ynyy
2
0λ
2
S + 81myn
2
yy
4
0λS − 54 f mynyy40λS − 54 f m2ynyy20λS, (B.12)
V
(1)
11 = Π
3
(
12 f 2y20m
2
yλ
2
S − 36 f 2y20m2yλS + 12 f 2m3yλ2S − 36 f 2m3yλS
− 6Δ f y40mynyλ2S + 18Δ f y40mynyλS − 6Δ f y20m2ynyλ2S
+ 18Δ f y20m
2
ynyλS − 18 f y40mynyλ2S + 54 f y40mynyλS − 4 f y20m2ynyλ3S
− 6 f y20m2ynyλ2S + 54 f y20m2ynyλS + 12Δy40myn2yλ2S − 54Δy40myn2yλS
+ 18y40myn
2
yλ
2
S − 54y40myn2yλS + 54Δy40myn2y
)
65
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
+Π5
(
36 f 2m3yλS − 12 f 2m3yλ2S − 12 f 2y20m2yλ2S
+ 36 f 2y20m
2
yλS + 6 f y
2
0m
2
ynyλ
2
S − 18 f y20m2ynyλS
)
, (B.13)
V
(0)
11 =
(
12 f mynyλ2Sy
2
0 + 3 fΔmynyλ
2
Sy
2
0 + 108 f mynyy
2
0 + 27 fΔmynyy
2
0
− 72 f mynyλSy20 − 18 fΔmynyλSy20
)
Π6 +
(−27Δ2n2yy40 + 81Δn2yy40
− 3Δ2n2yλ2Sy40 + 9Δn2yλ2Sy40 + 18Δ2n2yλSy40 − 54Δn2yλSy40
+ 7 f mynyλ3Sy
2
0 + fΔmynyλ
3
Sy
2
0 − 66 f mynyλ2Sy20 − 12 fΔmynyλ2Sy20
− 216 f mynyy20 − 54 fΔmynyy20 + 207 f mynyλSy20 + 45 fΔmynyλSy20
)
Π4
+
(
3Δn2yλ
3
Sy
4
0 −Δ2n2yλ3Sy40 + 27Δ2n2yy40 − 81Δn2yy40
− 81n2yy40 − 9 f 2λ2Sy40 + 9Δ2n2yλ2Sy40 − 27Δn2yλ2Sy40 − 18n2yλ2Sy40
+ 27 f nyλ2Sy
4
0 + 81 f nyy
4
0 + 27 f
2λSy
4
0 − 27Δ2n2yλSy40 + 81Δn2yλSy40
+ 81n2yλSy
4
0 − 108 f nyλSy40 + f mynyλ4Sy20 − 13 f mynyλ3Sy20
− fΔmynyλ3Sy20 − 18 f 2myλ2Sy20 + 90 f mynyλ2Sy20 + 9 fΔmynyλ2Sy20
+ 189 f mynyy20 + 27 fΔmynyy
2
0 + 54 f
2myλSy
2
0
− 243 f mynyλSy20 − 27 fΔmynyλSy20 − 9 f 2m2yλ2S + 27 f 2m2yλS
)
Π2
+ 81n2yy
4
0 − 81 f nyy40 − 3n2yy40λ3S + 3 f nyy40λ3S + 3 f mynyy20λ3S
− 81 f mynyy20 + 27n2yy40λ2S − 27 f nyy40λ2S − 27 f mynyy20λ2S
− 81n2yy40λS + 81 f nyy40λS + 81 f mynyy20λS, (B.14)
V
(9)
12 = 2Π
5y40myn
2
yλS − 2Π3y40myn2yλS (B.15)
V
(8)
12 = 3Π
4y40myn
2
yλS −Π2y40myn2yλS, (B.16)
V
(7)
12 = Π
5
(
12 f y20m
2
ynyλS − 18y40myn2yλS
)− 6 fΠ7y20m2ynyλS,
+Π3
(
20y40myn
2
yλS − 6 f y20m2ynyλS
)
, (B.17)
V
(6)
12 = −3 fΠ6y20m2ynyλS +Π4
(
12 f y40mynyλS + 18 f y
2
0m
2
ynyλS
− y40myn2yλ2S − 27y40myn2yλS
)
+Π2
(
6y40myn
2
yλ
2
S − 12 f y40mynyλS − 15 f y20m2ynyλS + 9y40myn2yλS
)
,
(B.18)
66
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
V
(5)
12 = Π
7(40 f y20m
2
ynyλS − 12 f 2m2yλS − 2 f y20m2ynyλ2S)
+Π5(12 f 2m2yλS + 6 f y
2
0m
2
ynyλ
2
S − 80 f y20m2ynyλS + 54y40myn2yλS)
+Π3
(
18 f y40mynyλS − 4 f 2m2yλS + 4 f y20myn2yλ2S − 4 f y20mynyλ2S
− 58 f y20myn2yλS + 58 f y20mynyλS + 2y40myn2yλ3S − 72y40myn2yλS
)
+ 4 f 2Π9m2yλS +Π(4y
4
0myn
2
yλ
2
S − 12 f y40mynyλS − 12 f y20m2ynyλS),
(B.19)
V
(4)
12 = Π
6( f y20m
2
ynyλ
2
S − 24 f 2m3yλS − 24 f 2y20m2yλS + 18 f y20m2ynyλS)
+Π4
(
48 f 2y20m
2
yλS + 48 f
2m3yλS − 72 f y40mynyλS
− 7 f y20m2ynyλ2S − 102 f y20m2ynyλS + 6y40myn2yλ2S + 81y40myn2yλS
)
+Π2
(
72 f y40mynyλS − 24 f 2y20m2yλS − 24 f 2m3yλS − 4 f y20myn2yλ2S
+ 4 f y20mynyλ
2
S − 84 f y20myn2yλS + 84 f y20mynyλS
+ 7y40myn
2
yλ
3
S − 36y40myn2yλ2S − 27y40myn2yλS
)
, (B.20)
V
(3)
12 = Π
7(36 f 2m2yλS − 8 f 2m2yλ2S + 10 f y20m2ynyλ2S − 78 f y20m2ynyλS)
+Π5
(
10 f 2m2yλ
2
S − 36 f 2m2yλS − 2 f y20m2ynyλ3S − 30 f y20m2ynyλ2S
+ 156 f y20m
2
ynyλS − 54y40myn2yλS
)
+Π3
(
36 f 2y40myλS
+ 72 f 2y20m
2
yλS − 4 f 2m2yλ2S + 36 f 2m3yλS + 12 f 2m2yλS
+ 6 f y40mynyλ
2
S − 108 f y40mynyλS + 2 f y20m2ynyλ3S
+ 26 f y20m
2
ynyλ
2
S − 186 f y20m2ynyλS − 6y40myn2yλ3S + 108y40myn2yλS
)
+Π
(−36 f 2y40myλS − 72 f 2y20m2yλS − 36 f 2m3yλS + 72 f y40mynyλS
+ 72 f y20m
2
ynyλS + 4y
4
0myn
2
yλ
3
S − 24y40myn2yλ2S
)
+Π9(2 f 2m2yλ
2
S − 12 f 2m2yλS),
(B.21)
V
(2)
12 = Π
6
(
72 f 2m3yλS − 12 f 2m3yλ2S − 12 f 2y20m2yλ2S + 72 f 2y20m2yλS
− 3 f y20m2ynyλ2S − 27 f y20m2ynyλS
)
+Π4
(
36 f 2y20m
2
yλ
2
S
− 144 f 2y20m2yλS + 36 f 2m3yλ2S − 144 f 2m3yλS + 108 f y40mynyλS
− 9 f y20m2ynyλ3S + 33 f y20m2ynyλ2S + 126 f y20m2ynyλS
− 9y40myn2yλ2S − 81y40myn2yλS
)
+Π2
(
72 f 2y20m
2
yλS
− 24 f 2y20m2yλ2S − 24 f 2m3yλ2S + 72 f 2m3yλS + 12 f y40mynyλ3S
− 108 f y40mynyλS − 21 f y20myn2yλ3S + 21 f y20mynyλ3S + 30 f y20myn2yλ2S
− 30 f y20mynyλ2S + 99 f y20myn2yλS − 99 f y20mynyλS
− 21y40myn2yλ3S + 54y40myn2yλ2S + 27y40myn2yλS
)
, (B.22)
67
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
V
(1)
12 = Π
3
(
108 f 2y40myλS − 18 f 2y40myλ2S − 36 f 2y20m2yλ2S + 216 f 2y20m2yλS
− 18 f 2m3yλ2S + 108 f 2m3yλS + 18 f y40mynyλ2S − 162 f y40mynyλS
− 4 f y20m2ynyλ3S + 42 f y20m2ynyλ2S − 198 f y20m2ynyλS + 54y40myn2yλS
)
+Π
(
36 f 2y40myλ
2
S − 108 f 2y40myλS + 72 f 2y20m2yλ2S − 216 f 2y20m2yλS
+ 36 f 2m3yλ
2
S − 108 f 2m3yλS − 12 f y40mynyλ3S + 108 f y40mynyλS
− 12 f y20m2ynyλ3S + 108 f y20m2ynyλS + 12y40myn2yλ3S − 36y40myn2yλ2S
)
+Π7(12 f y20m
2
ynyλ
2
S − 36 f y20m2ynyλS)+Π5
(
4 f y20m
2
ynyλ
3
S
− 36 f y20m2ynyλ2S + 72 f y20m2ynyλS
)
, (B.23)
V
(0)
12 = 2 fΠ
2y20myny(λS − 3)2λS
(
3Π2 + λS − 3
)
, (B.24)
V
(8)
22 =
(
1−Π2) y40myn2yλS [Π2 (3Δ− 2λS)+ 1] , (B.25)
V
(7)
22 = 2Π(1−Π2)y20mynyλS
[
f my + y
2
0ny (Δ− 2λS)
]
, (B.26)
V
(6)
22 = Π
4
(
5 f y20m
2
ynyλ
2
S − 6Δ f y20m2ynyλS −Δ2y40myn2yλS − 4Δy40myn2yλ2S
+ 30Δy40myn
2
yλS − y40myn2yλ3S − 9y40myn2yλ2S
)
+Π2
(
3Δ f y20m
2
ynyλS − 9 f y40mynyλS − 3 f y20m2ynyλ2S
− 9 f y20m2ynyλS +Δ2y40myn2yλS + 7Δy40myn2yλ2S − 30Δy40myn2yλS
− 4y40myn2yλ3S + 18y40myn2yλ2S + 12y40myn2yλS
)
+Π6(3Δ f y20m
2
ynyλS − 2 f y20m2ynyλ2S)
+ 9 f y40mynyλS + 9 f y
2
0m
2
ynyλS + y
4
0myn
2
yλ
2
S − 12y40myn2yλS, (B.27)
V
(5)
22 = Π
3
(
12 f y40mynyλ
2
S − 12 f 2y20m2yλS − 12 f 2m3yλS − 6Δ f y40mynyλS
− 6Δ f y20m2ynyλS + 14 f y20m2ynyλ2S + 18 f y20m2ynyλS − 2Δy40myn2yλ2S
+ 18Δy40myn
2
yλS − 2y40myn2yλ3S − 18y40myn2yλ2S
)
+Π
(
6 f 2y20m
2
yλS + 6 f
2m3yλS
+ 6Δ f y40mynyλS + 6Δ f y
2
0m
2
ynyλS − 12 f y40mynyλ2S − 8 f y20m2ynyλ2S
− 18 f y20m2ynyλS + 4Δy40myn2yλ2S − 18Δy40myn2yλS − 8y40myn2yλ3S
+ 36y40myn
2
yλ
2
S
)
+Π5(6 f 2m3yλS + 6 f
2y20m
2
yλS − 4 f y20m2ynyλ2S), (B.28)
68
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
V
(4)
22 = (4 f m
2
ynyy
2
0λ
3
S − 6 f m2ynyy20λ2S + 4 fΔm2ynyy20λ2S − 21 fΔm2ynyy20λS)Π6
+
(
6myn2yλ
3
Sy
4
0 −Δmyn2yλ3Sy40 − 2Δ2myn2yλ2Sy40 + 30Δmyn2yλ2Sy40
+ 9Δ2myn
2
yλSy
4
0 − 108Δmyn2yλSy40 + 2 f m2ynyλ3Sy20 − 15 f m2ynyλ2Sy20
− 11 fΔm2ynyλ2Sy20 + 42 fΔm2ynyλSy20
)
Π4 +
(
5Δmyn2yλ
3
Sy
4
0 − 2myn2yλ4Sy40
+ 22myn2yλ
3
Sy
4
0 + 3Δ
2myn
2
yλ
2
Sy
4
0 − 51Δmyn2yλ2Sy40 − 45myn2yλ2Sy40
− 6 f mynyλ2Sy40 − 9Δ2myn2yλSy40 + 108Δmyn2yλSy40
− 54myn2yλSy40 − 18 f 2myλSy40 + 81 f mynyλSy40 − 7 f m2ynyλ3Sy20
+ 15 f m2ynyλ
2
Sy
2
0 + 7 fΔm
2
ynyλ
2
Sy
2
0 − 36 f 2m2yλSy20 + 81 f m2ynyλSy20
− 21 fΔm2ynyλSy20 − 18 f 2m3yλS
)
Π2 − myn2yy40λ3S − 9myn2yy40λ2S
+ 15 f mynyy40λ
2
S + 15 f m
2
ynyy
2
0λ
2
S + 54myn
2
yy
4
0λS + 18 f
2myy
4
0λS
− 81 f mynyy40λS + 18 f 2m3yλS + 36 f 2m2yy20λS − 81 f m2ynyy20λS, (B.29)
V
(3)
22 = Π
3
(
72 f 2y20m
2
yλS − 18 f 2y20m2yλ2S − 18 f 2m3yλ2S + 72 f 2m3yλS
− 6Δ f y40mynyλ2S + 36Δ f y40mynyλS − 6Δ f y20m2ynyλ2S + 36Δ f y20m2ynyλS
− 6 f y40mynyλ3S − 18 f y40mynyλ2S − 4 f y20m2ynyλ3S − 12 f y20m2ynyλ2S − 54 f y20m2ynyλS
+ 12Δy40myn
2
yλ
2
S − 54Δy40myn2yλS + 12y40myn2yλ3S
)
+Π
(
12 f 2y20m
2
yλ
2
S
− 36 f 2y20m2yλS + 12 f 2m3yλ2S − 36 f 2m3yλS + 12Δ f y40mynyλ2S − 36Δ f y40mynyλS
+ 12Δ f y20m
2
ynyλ
2
S − 36Δ f y20m2ynyλS − 24 f y40mynyλ3S + 72 f y40mynyλ2S
− 22 f y20m2ynyλ3S + 48 f y20m2ynyλ2S + 54 f y20m2ynyλS + 2Δy40myn2yλ3S
− 24Δy40myn2yλ2S + 54Δy40myn2yλS − 4y40myn2yλ4S + 48y40myn2yλ3S
− 108y40myn2yλ2S
)
+Π5
(
6 f 2m3yλ
2
S − 36 f 2m3yλS + 6 f 2y20m2yλ2S − 36 f 2y20m2yλS
+ 2 f y20m
2
ynyλ
3
S + 6 f y
2
0m
2
ynyλ
2
S
)
, (B.30)
V
(2)
22 = (9 f
2m2yλ
2
S − 3 f 2m2yλ3S)Π8 +
(
6 f 2m2yλ
3
S − 12 f m2ynyy20λ3S
+ fΔm2ynyy
2
0λ
3
S − 18 f 2m2yλ2S + 36 f m2ynyy20λ2S − 18 fΔm2ynyy20λ2S
+ 45 fΔm2ynyy
2
0λS
)
Π6 +
(
6Δmyn
2
yλ
3
Sy
4
0 − 9myn2yλ3Sy40
−Δ2myn2yλ3Sy40 + 12Δ2myn2yλ2Sy40 − 72Δmyn2yλ2Sy40 + 27myn2yλ2Sy40
− 27Δ2myn2yλSy40 + 162Δmyn2yλSy40 + 3 f m2ynyλ4Sy20 − 12 f m2ynyλ3Sy20
− 6 fΔm2ynyλ3Sy20 + 9 f m2ynyλ2Sy20 + 48 fΔm2ynyλ2Sy20 − 90 fΔm2ynyλSy20
− 3 f 2m2yλ3S + 9 f 2m2yλ2S
)
Π4 +
(
Δmyn
2
yλ
4
Sy
4
0 + 6myn
2
yλ
4
Sy
4
0 + 3Δ
2myn
2
yλ
3
Sy
4
0
− 24Δmyn2yλ3Sy40 − 24myn2yλ3Sy40 − 15 f mynyλ3Sy40 − 18Δ2myn2yλ2Sy40
69
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
+ 117Δmyn
2
yλ
2
Sy
4
0 − 18 f 2myλ2Sy40 + 90 f mynyλ2Sy40 + 27Δ2myn2yλSy40
− 162Δmyn2yλSy40 + 108myn2yλSy40 + 108 f 2myλSy40 − 243 f mynyλSy40
− 5 f m2ynyλ4Sy20 + 15 f m2ynyλ3Sy20 + 5 fΔm2ynyλ3Sy20 − 36 f 2m2yλ2Sy20
+ 45 f m2ynyλ
2
Sy
2
0 − 30 fΔm2ynyλ2Sy20 + 216 f 2m2yλSy20 − 243 f m2ynyλSy20
+ 45 fΔm2ynyλSy
2
0 − 18 f 2m3yλ2S + 108 f 2m3yλS
)
Π2 − myn2yy40λ4S
+ 6myn2yy
4
0λ
3
S + 3 f mynyy
4
0λ
3
S + 3 f m
2
ynyy
2
0λ
3
S + 27myn
2
yy
4
0λ
2
S
+ 36 f 2myy
4
0λ
2
S − 90 f mynyy40λ2S + 36 f 2m3yλ2S + 72 f 2m2yy20λ2S
− 90 f m2ynyy20λ2S − 108myn2yy40λS − 108 f 2myy40λS + 243 f mynyy40λS
− 108 f 2m3yλS − 216 f 2m2yy20λS + 243 f m2ynyy20λS, (B.31)
V
(1)
22 =
(
72 f 2λ2Sm
3
y − 18 f 2λ3Sm3y − 54 f 2λSm3y − 18 f 2y20λ3Sm2y + 6 f nyy20λ3Sm2y
+ 72 f 2y20λ
2
Sm
2
y − 18 f nyy20λ2Sm2y − 54 f 2y20λSm2y
)
Π5 +
(
18myn2yλ
3
Sy
4
0
− 18 f mynyλ3Sy40 + 18Δmyn2yλ2Sy40 − 54myn2yλ2Sy40 + 54 f mynyλ2Sy40
− 18 fΔmynyλ2Sy40 − 54Δmyn2yλSy40 + 54 fΔmynyλSy40 − 4 f m2ynyλ4Sy20
+ 24 f 2m2yλ
3
Sy
2
0 − 12 f m2ynyλ3Sy20 − 108 f 2m2yλ2Sy20 + 90 f m2ynyλ2Sy20
− 18 fΔm2ynyλ2Sy20 + 108 f 2m2yλSy20 − 54 f m2ynyλSy20 + 54 fΔm2ynyλSy20
+ 24 f 2m3yλ
3
S − 108 f 2m3yλ2S + 108 f 2m3yλS
)
Π3 +
(
12 f mynyλ4Sy
4
0
− 12myn2yλ4Sy40 + 6Δmyn2yλ3Sy40 + 72myn2yλ3Sy40 − 72 f mynyλ3Sy40
− 6 fΔmynyλ3Sy40 − 36Δmyn2yλ2Sy40 − 108myn2yλ2Sy40 + 108 f mynyλ2Sy40
+ 36 fΔmynyλ2Sy
4
0 + 54Δmyn
2
yλSy
4
0 − 54 fΔmynyλSy40 + 12 f y2mynyλ4Sy20
+ 12 f mynyλ
4
Sy
2
0 − 24 f ymynyλ4Sy20 − 6 f 2m2yλ3Sy20 − 66 f y2mynyλ3Sy20
− 66 f mynyλ3Sy20 + 132 f ymynyλ3Sy20 − 6 f y2Δmynyλ3Sy20 − 6 fΔmynyλ3Sy20
+ 12 f yΔmynyλ
3
Sy
2
0 + 36 f
2m2yλ
2
Sy
2
0 + 72 f y
2mynyλ
2
Sy
2
0 + 72 f mynyλ
2
Sy
2
0
− 144 f ymynyλ2Sy20 + 36 f y2Δmynyλ2Sy20 + 36 fΔmynyλ2Sy20
− 72 f yΔmynyλ2Sy20 − 54 f 2m2yλSy20 + 54 f y2mynyλSy20 + 54 f mynyλSy20
− 108 f ymynyλSy20 − 54 f y2ΔmynyλSy20 − 54 fΔmynyλSy20
+ 108 f yΔmynyλSy20 − 6 f 2m3yλ3S + 36 f 2m3yλ2S − 54 f 2m3yλS
)
Π, (B.32)
V
(0)
22 = (18 fΔmynyy
2
0λ
2
S − 3 fΔmynyy20λ3S − 27 fΔmynyy20λS)Π6 +
(
3Δ2n2yλ
3
Sy
4
0
− 9Δn2yλ3Sy40 − 18Δ2n2yλ2Sy40 + 54Δn2yλ2Sy40 + 27Δ2n2yλSy40 − 81Δn2yλSy40
− 3 f mynyλ4Sy20 − fΔmynyλ4Sy20 + 18 f mynyλ3Sy20 + 12 fΔmynyλ3Sy20
70
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
− 27 f mynyλ2Sy20 − 45 fΔmynyλ2Sy20 + 54 fΔmynyλSy20
)
Π4
+
(
Δ2n2yy
4
0λ
4
S − f mynyy20λ5S − 3Δn2yy40λ4S + 9 f mynyy20λ4S + fΔmynyy20λ4S
− 27 f 2y40λ3S − 9Δ2n2yy40λ3S + 27Δn2yy40λ3S − 18n2yy40λ3S + 45 f nyy40λ3S
− 27 f 2m2yλ3S − 54 f 2myy20λ3S + 18 f mynyy20λ3S − 9 fΔmynyy20λ3S + 135 f 2y40λ2S
+ 27Δ2n2yy
4
0λ
2
S − 81Δn2yy40λ2S + 81n2yy40λ2S − 216 f nyy40λ2S + 135 f 2m2yλ2S + 270 f 2myy20λ2S
− 189 f mynyy20λ2S + 27 fΔmynyy20λ2S − 162 f 2y40λS − 27Δ2n2yy40λS + 81Δn2yy40λS
− 81n2yy40λS + 243 f nyy40λS − 162 f 2m2yλS − 324 f 2myy20λS + 243 f mynyy20λS
− 27 fΔmynyy20λS
)
Π2 + 3n2yy
4
0λ
4
S − 3 f nyy40λ4S − 3 f mynyy20λ4S + 18 f 2y40λ3S − 9n2yy40λ3S
− 9 f nyy40λ3S + 18 f 2m2yλ3S + 36 f 2myy20λ3S − 9 f mynyy20λ3S − 108 f 2y40λ2S − 27n2yy40λ2S
+ 135 f nyy40λ
2
S − 108 f 2m2yλ2S − 216 f 2myy20λ2S + 135 f mynyy20λ2S + 162 f 2y40λS
+ 81n2yy
4
0λS − 243 f nyy40λS + 162 f 2m2yλS + 324 f 2myy20λS − 243 f mynyy20λS. (B.33)
Appendix C. Wormholes with toroidal boundaries in simple low-energy
theories
This appendix collects some results regarding wormholes with toroidal boundaries. As in the
spherical case, we consider both the U(1)3 theory of section 4 (see appendix C.1) and a scalar
theory (see appendix C.2), though now the latter will contain three complex scalars. These
results are less complete than for the spherical-boundary cases of sections 4 and 5, in part
because we construct only wormhole solutions and do not construct a disconnected solution.
Indeed, in the torus case a smooth disconnected solution must feature a preferred cycle of the
torus that shrinks to zero size while the other cycles remain finite (much as in the familiar AdS
soliton [66, 67]). This requires a metric ansatz that breaks the discrete symmetries we impose
below.
Despite this lack of completeness, the results below indicate that toroidal solutions are
broadly similar to thosewith spherical boundaries. In particular, in the scalar case we again find
a Hawking–Page-like structure for the wormhole phases, and in particular the large wormhole
branch is stable. In contrast, in the U(1)3 case we identify only a single branch of wormhole
solutions which we find exists for arbitrarily small values of appropriate boundary sources.
C.1. U(1)3 with a toroidal boundary
As a short asidewemention that there is a very simple four-dimensionalexample of awormhole
with three gauge fields when the boundarymetric is a torus.We take the same theory as in (4.1)
but with three gauge fields of the form
AI = LB0
εIJK
2
xJ dxK (C.1)
71
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
with B0 constant, and a metric of the form
ds2 =
dr2
f
+ (r2 + r20)
(
dx21 + dx
2
2 + dx
2
3
)
. (C.2)
A solution exists provided
f (r) =
r2 + 2r20
L2
and B0 =
r20
L2
. (C.3)
In this section we are assuming that x1, x2 and x3 are periodic coordinates with periodsΔx1,
Δx2, Δx3. In this context we believe that the solution we have found is the unique connected
geometry, though we have not found a way to construct a disconnected solution. It is a simple
exercise to compute the regulated on-shell action for this solution which yields
S =
8
√
2K
(
1
2
)
r30
L
Δx1Δx2Δx3. (C.4)
We have not attempted to study the negative modes of this solution, though we believe that
this wormhole will again be stable. The reason for this is that the infinite-radius limit of the
U(1)3 wormholes with a spherical boundary we constructed are connected coincides with the
Δxi →∞ limit of the wormholes discussed in this section. Since for the spherical wormholes
we found no negative modes, we expect the same to hold here.
C.2. Wormholes sourced by scalar fields with a toroidal boundary
We now consider wormholes sourced by scalar fields which have a toroidal boundary. If we
want to keep isotropy and homogeneity, we seem to need at least three complex scalar fields,
which we label by ψI and collectively assemble in a vector ψ. We will proceed much as in
section 5, using the simple low-energy action
S = −
∫
M
d4x
√
g
[
R+
6
L2
− 2(∇a ψ) · (∇a ψ)∗ − 2μ2 ψ · ψ∗
]
− 2
∫
∂M
d3x
√
hK + Sμ
2
B . (C.5)
Here L is the four-dimensionalAdS length scale and ∗ denotes complex conjugation, the second
term is the usual Gibbons–Hawking term and Sμ
2
B the boundary counter-term tomake the action
finite and the variational problem well defined from the perspective of ψ. We again consider
both the massless example and the effective mass that would describe conformal coupling, so
the boundary terms Sμ
2
B are chosen as in section 5 (but now with three complex scalar fields).
The Einstein equation and scalar field equation (ignoring boundary terms) derived from this
action read
Rab − R2 gab −
3
L2
gab = 2∇(a ψ · ∇b) ψ∗ − gab∇c ψ · ∇c ψ∗ − μ2gab ψ · ψ∗,
(C.6a)
ψ = μ2 ψ. (C.6b)
72
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Before describing our (numerical) solutionswith torus boundary,we would like to comment
on a simple analytic wormhole that arises for the massless case when the torus boundary is
replaced by R3. For this solution one considers the configuration
ψ =
⎡
⎣Φ0x1Φ0x2
Φ0x3
⎤
⎦ , (C.7a)
with metric
ds2 =
dr2
f (r)
+ (r2 + r20)(dx
1
1 + dx
2
1 + dx
3
1), (C.7b)
taking r0 = Φ0L and
f =
r2 + r20
L2
. (C.7c)
This shows rather explicitly that wormhole solutions can exist for any value of A0, though the
solution does not satisfy standard boundary conditions at large xi.
To fix this, we introduce harmonic dependence on xi in the scalar field ansatz and also
consider μ 
= 0 to write
ψ = Xkψ(r), (C.8)
with
Xk =
⎡
⎣eikx1eikx2
eikx3
⎤
⎦ . (C.9)
We also consider the metric
ds2 =
dr2
f (r)
+ (r2 + r20)(dx
2
1 + dx
2
2 + dx
2
3), (C.10)
where f , ψ and r0 are to be determined numerically for a given source Φ0 associated with the
boundary value of ψ. In performing such numerics it is wise to use a compact coordinate, so
we introduce
y = 1− r0√
r2 + r20
. (C.11)
The conformal boundary is now located at y = 1, and the Z2 symmetry plane at y = 0.
We are interested in the case where x1, x2 and x3 span a cubic three-torus T3 with period
 = 2π/k. By construction, the torus has minimal volume at r = y = 0. It is a simple exercise
to determine f from the Einstein equation to find
f =
L2
[
k2(1− y)2 + r20μ2
]
ψ2 − r20
L2(2− y)(1− y)2y [(1− y)2ψ′2 − 1] . (C.12)
73
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Since we are interested in solutions for which ψ is smooth at y = 0 and f is finite there, we
need to have
L2(k2 + r20μ
2)ψ(0)2 = r20. (C.13)
At this stage we introduce ψ(0) = A0 and write r0 in terms of A0 in the equation for ψ. This
yields
m2yψ
′′ +
p(0,1,1)y
ψ
+ myp
(2,1,2)
y ψ
′ − m2y p(0,1,1)
ψ′2
ψ
− m3y p(3,2,3)y ψ′3 = 0, (C.14a)
where
p(a,b,c)y =
aA20 − (bm2y + cA20L2μ2)ψ2
A20 − (m2y + A20L2μ2)ψ2
, (C.14b)
and we again recall that my = 1− y. The equation for ψ depends only on A0, and the depen-
dence in r0 and k1 has dropped out. This is to be expected. If the boundary metric is flat, there
is a residual gauge freedom when one simultaneously scales all the xi and uses conformal
invariance. A priori one might have thought that the wormholes we seek to construct formed
a two-dimensional family of solutions parametrised by Φ0 and k with r0 being fixed by the
former. However, due to conformal invariance this is not the case, and instead only the ratio
V/kΔ− is physically meaningful in the bulk. One might erroneously think that this should
have reduced the moduli space of solutions in the spherical case to 0 dimensions. However, the
sphere there introduces a new scale in the problem which cannot be removed.
Since we do not construct a disconnected solution for comparison, computing the on-shell
action is not of much interest. Instead, we will focus on trying to understand whether worm-
holes in this class of theories exist to arbitrary small values of Φ0 and whether they are free of
negative modes. The answer to the first question appears to be negative. Even for wormholes
with toroidal boundary conditions we find a minimal critical amplitude V/kΔ− above which
they can exist. This is perhaps surprising, as we find no smooth solutions at all within our
ansatz for V/k less than this critical value. For the massless case we find that wormholes only
exist for V  Vmin ≈ 1.7107(3) (see left-hand side of figure 15), whereas for the conformal
case we need V/k > (V/k)min ≈ 11.2529(7) (see right-hand side of figure 15). Just as for the
spherical case, for each value of V  Vmin two wormhole solutions exist. We again call the
phase with smallest y0/k (with y0 ≡ r0/L) the small wormhole phase and we call the phase
with largest y0/k the large wormhole phase. Precisely at V/k
Δ−
min we have y0/k = (y0/k)min
with (y0/k)min ≈ 1.10146(5) and (y0/k)min ≈ 0.49202(4) for themassless and conformal cases,
respectively. The small wormhole phase is shown in figure 15 as orange squares, while the large
wormhole phase is represented by the blue disks.
For the massless case we can actually do better and find a uniform expansion for ψ in
powers of 1/A0. This turns out to be a very useful expansion because this allows us to check
our numerics. The expansion takes the rather simple form
ψ = A0 +
+∞∑
i=0
1
A2i+10
ψ(i)
(
r
r0
)
. (C.15)
We have carried out this expansion to i = 8, but for the sake of brevity we only present here
results for i = 0, 1
ψ(0)(z) =
1
2
z2
1+ z2
, (C.16)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 15. Wormholes with toroidal boundary conditions. Left panel: wormholes
sourced by massless scalars, with the dashed red line being analytically generated by
equation (C.15) with terms up to i = 8. Right panel: wormholes generated by confor-
mally coupled scalars. In both panels, the small wormhole phase is represented by orange
squares, while large wormholes are given by blue disks.
ψ(1)(z) =
1
48
[
z2
(
9+ 27z2 + 14z4
)
(
1+ z2
)3 − 6z1+ z2 arctan z− 3 arctan2
]
. (C.17)
On the left panel of figure 15we compare our analytic expansion up to i = 8, with the numerical
data and find excellent agreement for a large range of A0.
C.2.1. Negative modes with toroidal boundaries. Studying perturbations of wormholes with
planar boundaries turns out to be schematically similar to studying those with spherical bound-
aries, though it is easier in practice. Again, we expand all our perturbations in terms of scalar,
vector and tensor harmonics on T3 obeying to
T3SkS + k2SSkS = 0, (C.18a)

T3S
kV
i + k
2
VS
kV
i = 0 with ∇ iSkVi = 0, (C.18b)
and
T3SkTi j + k2TS
kT
i = 0, with ∇ iSkTi j = 0 and 𝕘i jSkTi j = 0, (C.18c)
respectively.Modeswith kS = kV = 0 have to be studied separately, but kT = 0 can be obtained
from the result with kT = 0.
C.2.1.1. Scalar-derived perturbations with kS = 0. This section is similar in many respects to
the section where we studied the negative mode of a spherically symmetric wormhole with
respect to scalar-derived perturbations with  = 0. We shall see that the large wormhole phase
has no negative modes, whereas the small wormhole phase does seem to possess such a mode.
75
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Our perturbations read
δ ds2 =
L2
(1− y)2
[
δ f (y)
dy2
(2− y)y + δp(y)(dx
2
1 + dx
2
2 + dx
2
3)
]
, (C.19a)
and for the scalars
δ ψ = Xkδψ. (C.19b)
Under an infinitesimal diffeomorphism ξ = ξy dy these perturbations transform as
δ f =
2(1− y)
L2
[
1− 4y+ 2y2 − y(2− y)(1− y) f
′
2 f
]
ξy +
2y(2− y)(1− y)2
L2
ξ′y,
(C.20a)
δp=
2(2− y)(1− y)yy20
L2 f
ξy, (C.20b)
δψ =
(2− y)(1− y)2yψ′
L2 f
ξy. (C.20c)
By now the procedure is familiar. We first expand the action (C.5) to second order in the
perturbations. The resulting action, S(2) is a function of δψ, δ f and their first derivatives with
respect to y. Additionally, S(2) is also a function of δp and its first and second derivatives. We
first integrate by parts the term proportional to δp′′ with the resulting boundary term canceling
the Gibbons–Hawking–York perturbed boundary action. S(2) is now a function of δψ, δ f , δp
and their first derivatives. One can also integrate one more my parts terms proportional to δ f ′
(whose boundary terms cancel with the perturbed boundary counter terms appearing in (C.5)).
The second order action S(2) is then a function δψ, δp and their first derivatives and of δ f .
Crucially, δ f enters the action algebraically. This means we can perform the Gaussian integral
over δ f (again using the Wick rotation described in section 3) and find an effective action Sˇ(2)
that is a function of δψ, δp and their first derivatives only. At this stage we introduce the gauge
invariant quantity
Q =
√
6
[
δψ − (1− y)ψ
′
2y20
δp
]
, (C.21)
where the factor of
√
6 was chosen for later convenience of presentation. ClearlyQ is invariant
under the infinitesimal gauge transformations (C.20). Solving the above relation with respect
to δψ—gives an action for Q, where the dependence in δp completely drops out because of
gauge invariance. The final action for Q reads
Sˇ(2) = 2L23
∫ +∞
0
dy
y30
√
f√
2− y(1− y)4√y
[
(2− y)(1− y)2y
1− (1− y)2ψ′2
Q′2
f
+ VQ2
]
,
(C.22)
where
V =
1
ψ2
{
(2− y)y
f
+
2 f
[
y20 − k2(1− y)3ψ3ψ′
]
(2− y)yy20
[
1− (1− y)2ψ′2]2 −
3
1− (1− y)2ψ′2
}
. (C.23)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
As expected, V is not positive definite for all wormholes. The combination 1 − (1 − y)2ψ′2
which appears multiplying the kinetic term for Q is positive definite so long as y0/k = 1
15 and
y0/k  0.4529(0), for the massless and conformally coupled cases, respectively. In particular,
for the large wormhole branch 1 − (1 − y)2ψ′2 is positive definite.
To proceed, we use numerical methods. We first note that in the original r coordinates of
(C.7b), V would have been even around r = 0. This means perturbations that are even and odd
with respect to r = 0 will be orthogonal, so we can study them separately. In terms of the y
coordinates, these correspond to very distinct behaviors near y = 0. Namely, in the odd sector
we have Q ∼ √y near the origin, while in the even sector Q admits a regular Taylor expansion
around y = 0. We have not found any negative mode on the odd sector of perturbations.
To search for negative modes λ, we integrate (C.7b) by parts and set
−
√
2− y(1− y)4√y√
f
[
1
(1− y)2
√
2− y√y
1− (1− y)2ψ′2
Q′√
f
]′
+ VQ = λQ. (C.24)
Near the boundary, we find that admits two possible boundary behaviors
Q = C+(1− y)
3
2+
√(
Δ− 32
)2−λ
[1+ · · · ]+ C−(1− y)
3
2−
√(
Δ− 32
)2−λ
[1+ · · · ] , (C.25)
with normalisability demanding we set C− = 0. We thus have a well defined Sturm–Liouville
problem, which we can readily solve numerically.
The results of this analysis can be seen in figure 16 where we plot λ as a function of y0/ymin0
for the massless (left panel) and conformal (right panel) cases. In both cases, a negative mode
exists for the small wormhole phase, but becomes positive on the large wormhole phase. This
establishes that large wormholes are stable with respect to scalar-derived perturbations with
kS = 0.
C.2.1.2. Scalar-derived perturbations with kS 
= 0. We have chosen our boundary metric to
be a torus so our fundamental scalar harmonic takes a very simple form
S
kS = cos(kS · x+ γS), (C.26)
with kS = |kS| and x = {x1, x2, x3}. Note also that kS = k{n1, n2, n3}, where ni are integers
because our three-torus is cubic. Finally, γS is an arbitrary phase which will play no role.
While we were able to keep all ni distinct, it is clear the most dangerous sector occurs when
we take all ni = n. This is the sector we present here. We also define kS =
√
3κ, with κ = nk.
The metric perturbations take the already familiar form
δ ds2kS = h
kS
yy(y)S
kS dy2 + 2hkSy (y)∇ iSkS dy dxi
+ HkST (y)S
kS
i j dx
i dx j + HkSL (y)S
kS𝕘i j dxi dx j, (C.27a)
where
S
kS
i j = ∇ jSkS −
𝕘i j
3
∇ kSkS , (C.27b)
while for the scalar perturbation we choose
15 This can be analytically proved by manipulating the scalar equation (C.14a) and numerically checked to be the case
for the first 100 digits.
77
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Figure 16. Negative mode for the k = 0 perturbations as a function of y0/ymin0 . Left
panel: negative modes of wormholes sourced by massless scalars. Right panel: negative
modes of wormholes generated by conformally coupled scalars.
δ ψkS =
XkBkS(y)S
kS + (∇ iSkS∇ iXk)AkS (y). (C.27c)
In the above 𝕘 is the metric on T3 and∇ its associated metric preserving connection.
Under an infinitesimal diffeomorphism of the form
ξkS = ξkSy (y)dy+ L
kS
y (y)∇ iSkS dxi (C.28)
the metric and scalar perturbations transform as
δAkS =
(1− y)2ψ
L2y20
LkSy , (C.29a)
δBkS =
(2− y)y(1− y)2ψ′
L2 f
ξkSy , (C.29b)
δhkSyy =
2
(
1− 4y+ 2y2)
y(2− y)(1− y) ξ
kS
y −
f ′
f
ξkSy + 2ξ
kS′
y , (C.29c)
δhkSy = ξ
kS
y + L
kS′
y − 2
1− yL
kS
y , (C.29d)
δHkST = 2L
kS
y , (C.29e)
δHkSL =
2(2− y)yy20
(1− y) f ξ
kS
y − 2κ2LkSy . (C.29f)
Had we taken all ni distinct, we would have to consider three different perturbations similar to
AkS and BkS , parametrising each of the complex scalars in
ψ.
The remaining procedure is very similar to what we have seen when studying negative
modes of the wormholes sourced by the scalars with a spherical boundary. First, we write the
second order action S(2) in first order form by integrating by parts and find it can be written in
term of AkS , BkS ,H
kS
L ,H
kS
T , h
kS
y and their first derivatives. However, h
kS
yy appears algebraically and
78
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
we again apply the Wick-rotation procedure of section 3. We can thus perform the Gaussian
integral and find a new second order action S˜(2) that is a function of AkS , BkS , H
kS
L , H
kS
T and their
first derivatives, but now hkSy enters algebraically and again we can perform the corresponding
Gaussian integral finding an action Sˇ(2) that is a function of AkS , BkS , H
kS
L , H
kS
T and their first
derivatives.
At this point we introduce gauge invariant variables QkS1 and Q
kS
2 defined by
QkS1 = AkS −
(1− y)2ψ
2L2y20
HkST , (C.30a)
QkS2 = BkS −
(1− y)3κ2ψ′
2L2y20
HkST −
(1− y)3ψ′
2L2y20
HkSL . (C.30b)
which are invariant under the infinitesimal transformations (C.29). Solving the above relations
with respect to AkS and BkS and inputting those in Sˇ
(2) gives an action for QkS1 , Q
kS
2 and their
first derivatives only. The dependence in HkST and H
kS
L , after using the equations of motion for
ψ, completely drops out by virtue of gauge invariance. To ease presentation we define further
QkS1 =
1
2
√
6π3/2
√
k
mΔy ψ
[
qkS1 −
(1− y)2
n2
ψ′qkS2
]
, (C.31)
QkS2 =
1
2
√
6π3/2
√
k
mΔy q
kS
2 . (C.32)
The second order action Sˇ(2) can then be written as
Sˇ(2) = 2L2
∫ +∞
0
dy
y30m
2Δ−4
y
√
f√
ny
[
m2yny
f
q
kS ′
i K
i jq
kS ′
j + q
kS
i V
i jqkSj
]
, (C.33)
where
K
−1 =
⎡
⎢⎣
1
n4
(
1+
n2
ψ2
+ 3m2yψ
′2
)
2myψ′
n2
2myψ′
n2
1
⎤
⎥⎦ . (C.34)
It is a simple exercise to show thatK is positive definite so long as 1 − (1− y)2ψ′2 is positive.
However, we have argued that this is the case for all large wormholes. It then all boils down to
the positivity properties of V, whose explicit form we present in appendix C.2.1. This easy to
study numerically, and we find that, for |n| = 116, V is positive definite for y0/k  0.4173(5)
and y0/k  0.9207(9) for the conformal and massless cases, respectively. Both these values
are well within the small wormhole branch, so these results establish stability in the large
wormhole branch.
C.2.1.2.1. Symmetric matrices for the scalars with toroidal boundary conditions
The symmetric matrix V is given by
VIJ = (K−1)IK(K−1)JMVKM. (C.35)
16 Other values of |n| > 1 are even more stable.
79
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
We then have
VIJ =
1
A20 fψ
2dIJ
6∑
i=0
ψ′iV(i)IJ (C.36)
with V(6)12 = V
(6)
22 = V
(5)
12 = 0, ny = y(2− y), my = 1− y and
d11 = A
2
0n
4ψny, (C.37)
d12 = A
2
0n
2ny, (C.38)
d22 = 1. (C.39)
Furthermore
V
(6)
11 = 72A
4
0ψ
3m6yn
2
y , (C.40)
V
(5)
11 = 48A
4
0ψ
2m5yn
2
y , (C.41)
V
(4)
11 = A
4
0
(
42Δ2 fψ5m6yny − 126Δ fψ5m6yny − 27Δψ3m4yn2y − 72ψ3m4yn2y + 4ψm4yn2y
)
− 42A20 fψ5m6yny, (C.42)
V
(3)
11 = A
4
0
(
16Δ2 fψ4m5yny − 48Δ fψ4m5yny + 48 fψ2m3yny − 6Δψ2m3yn2y − 48ψ2m3yn2y
)
− 16A20 fψ4m5yny, (C.43)
V
(2)
11 = A
2
0
(
36Δ f 2ψ7m6y − 12Δ2 f 2ψ7m6y + 9 f n2ψ3m4yny + 9Δ fψ5m4yny + 14 fψ5m4yny
)
+ A40
(
6Δ4 f 2ψ7m6y − 36Δ3 f 2ψ7m6y + 54Δ2 f 2ψ7m6y − 9Δ2 f n2ψ3m4yny
+ 27Δ f n2ψ3m4yny − 9Δ3 fψ5m4yny + 13Δ2 fψ5m4yny + 42Δ fψ5m4yny + 4 fψm2yny
− 3Δn2ψm2yn2y − 3Δ2ψ3m2yn2y + 24Δψ3m2yn2y − 4ψm2yn2y
)
+ 6 f 2ψ7m6y , (C.44)
V
(1)
11 = A
4
0
(
6Δ2 f 2ψ4m3y − 18Δ f 2ψ4m3y + 4Δ2 f n2ψ2m3yny − 12Δ f n2ψ2m3yny
− 6Δ fψ2myny + 2Δn2myn2y + 6Δψ2myn2y
)− A20 (6 f 2ψ4m3y + 4 f n2ψ2m3yny) ,
(C.45)
V
(0)
11 = A
4
0
(
21Δ f n2ψ3m2yny −Δ3 f n2ψ3m2yny − 4Δ2 f n2ψ3m2yny − 3Δ2 f n4ψm2yny
+ 9Δ f n4ψm2yny +Δ
3(− f )ψ5m2yny −Δ2 fψ5m2yny + 12Δ fψ5m2yny −Δ2n2ψn2y
+ 3Δn2ψn2y −Δ2ψ3n2y + 3Δψ3n2y
)
+ A20
(
Δ f n2ψ3m2yny + 7 f n
2ψ3m2yny
+ 3 f n4ψm2yny +Δ fψ
5m2yny + 4 fψ
5m2yny
)
, (C.46)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
V
(5)
12 = 36A
4
0ψ
2m5yn
2
y , (C.47)
V
(4)
12 = 24A
4
0ψm
4
yn
2
y , (C.48)
V
(3)
12 = A
4
0
(
18Δ2 fψ4m5yny − 54Δ fψ4m5yny − 12Δψ2m3yn2y − 36ψ2m3yn2y + 2m3yn2y
)
− 18A20 fψ4m5yny, (C.49)
V
(2)
12 = A
4
0
(
8Δ2 fψ3m4yny − 24Δ fψ3m4yny + 24 fψm2yny − 2Δψm2yn2y − 24ψm2yn2y
)
− 8A20 fψ3m4yny, (C.50)
V
(1)
12 = A
2
0(12Δ f
2ψ6m5y − 4Δ2 f 2ψ6m5y + 6 f n2ψ2m3yny + 4Δ fψ4m3yny + 6 fψ4m3yny)
+ A40
(
2Δ4 f 2ψ6m5y − 12Δ3 f 2ψ6m5y + 18Δ2 f 2ψ6m5y − 6Δ2 f n2ψ2m3yny
+ 18Δ f n2ψ2m3yny − 4Δ3 fψ4m3yny + 6Δ2 fψ4m3yny + 18Δ fψ4m3yny + 2 f myny
− 2Δ2ψ2myn2y + 12Δψ2myn2y − 2myn2y
)
+ 2 f 2ψ6m5y , (C.51)
V
(0)
12 = A
4
0
(
2Δ2 f 2ψ3m2y − 6Δ f 2ψ3m2y + 2Δ2 f n2ψm2yny − 6Δ f n2ψm2yny
− 2Δ fψny + 2Δψn2y
)− A20 (2 f 2ψ3m2y + 2 f n2ψm2yny) (C.52)
V
(4)
22 = 18A
2
0ψ
2m4yny, (C.53)
V
(3)
22 = 12A
2
0ψm
3
yny, (C.54)
V
(2)
22 = A
2
0(6Δ
2 fψ4m4y − 18Δ fψ4m4y − 3Δψ2m2yny − 18ψ2m2yny + m2yny)− 6 fψ4m4y ,
(C.55)
V
(1)
22 = A
2
0(4Δ
2 fψ3m3y − 12Δ fψ3m3y + 12 fψmy − 12ψmyny)− 4 fψ3m3y (C.56)
V
(0)
22 = A
2
0
(
9Δ f n2ψ2m2y − 3Δ2 f n2ψ2m2y +Δ3(− f )ψ4m2y + 3Δ2 fψ4m2y + f −Δ2ψ2ny
+ 3Δψ2ny − ny
)
+ 3 f n2ψ2m2y +Δ fψ
4m2y . (C.57)
C.2.1.3. Vector-derived perturbations with kV = 0. This mode is special, and must be studied
separately. It is the analogue of the mode with V = 1 in the spherical case. Again we find that
the associated S(2) vanishes identically after integrating out hy (whose quadratic coefficient has
the correct sign to make the integral over hy converge). Thus the other parts of this mode are
pure-gauge.
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
C.2.1.4. Vector-derived perturbations with kV 
= 0. Vector derived perturbations with kV 
= 0
follow a similar pattern to the spherical case. Recall that vector harmonics on T3 must be
transverse, i.e.∇ iSi = 0 and obey to
T3S
kV
i + k
2
VS
kV
i = 0. (C.58)
One such example is for instance
S
kV
i dx
i = cos(kV x1 + γ)dx3, (C.59)
where γ is an unimportant phase and kV = nk. From S
kV
i we can construct the following
symmetric tensor
S
kV
i j = ∇ iSkVj +∇ jSkVi , (C.60)
which we use to build the most general vector-derived metric perturbation. Vector-derived
perturbation with kV read
δ ds2kV = 2h
kV
y (y)S
kV
i dy dx
i + HkVT (y)S
kV
i j dx
i dx j (C.61a)
while for the scalar perturbation we choose
δΠkV = (S
kV
i ∇ iXk)AkV (y). (C.61b)
The most general vector-derived infinitesimal diffeomorphism can be constructed via
ξV = LkVy S
kV
i dx
i. (C.62)
The above infinitesimal diffeomorphism induces the following gauge transformations
δhkVy = −
2LkVy
1− y + L
kV
y
′
, (C.63a)
δHkVT = L
V
y , (C.63b)
δAkV =
(1− y)2ψ
L2y20
LkVy . (C.63c)
By now the procedure should be very familiar. We expand the action to second order in
perturbations, and write it in first order form. To do this, we have to integrate by parts, and the
boundary terms cancel with the perturbed Gibbons–Hawking–York term. It turns out that hkSy
enters the second order action S(2) algebraically, which means we can do the Gaussian integral
and find a new action Sˇ(2) which is a function of HkST and AkS only. At this stage we introduce
a gauge invariant variable QkV defined through the relation
AkV =
√
kmΔy
2
√
2π3/2
√
1+
4ψ2
n2
QkV +
(1− y)2ψ
L2y20
HkVT (C.64)
which leads to the following second order action
Sˇ(2) = 2L2
∫ ∞
0
dy
√
f y30m
2(Δ−2)
y√
ny
[
m2yny
f
Q′kV
2
+ VQ2kV
]
(C.65)
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Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
with
V = m2y
(
1
A20
− μ2L2
)(
n2 + 4ψ2 +Δψ2
)− nyμ2L2
f
+
n2(
n2 + 4ψ2
)
ψ2
− ny
f
{
3Δm2yψ
′2 +
1(
n2 + 4ψ2
)
ψ2
[
n2 − m
2
yn
2
(
n2 − 8ψ2)ψ′2
n2 + 4ψ2
]}
,
(C.66)
where we recall that ny ≡ y(2− y), my ≡ 1− y and kV = nk. Though it is not apparent from
the above expression, it turns out that V seems positive definite for all wormholes we have
constructed. In particular, it appears positive for the largewormholes, thus establishing stability
with respect to vector-derived perturbations in this sector as well.
C.2.1.5. Tensor-derived perturbations. We now come to the easiest sector of perturbations.
The building blocks for perturbations in this sector are given by tensor harmonics on T3, which
obey to
T3SkTi j + k2T S
kT
i j = 0 (C.67)
with∇ iSkTi j = 0 and 𝕘i jSkTi j = 0. An example of such an harmonic is
S
kT
i j dx
i dx j = cos(kT x1 + γ)dx2 dx3, (C.68)
where γ is again an arbitrary phase and kT = nk.
The metric perturbation is simply given by
δ ds2kT =
√
2k3/2L2y20
π3/2m2y
HkT (y)S
kT
i j dx
i dx j, (C.69)
while for ψ we demand δψ = 0. In this caseHkT is automatically gauge invariant, and bringing
the quadratic action to first order form yields
Sˇ(2) = 2L2
∫ +∞
0
dy
√
f y30
m4y
√
ny
[
m2yny
f
H′kT
2
+ VH2kT
]
(C.70)
with
V =
k2m2y
y20
(
n2 + 4ψ2
)
, (C.71)
which is manifestly positive. This thus establishes that no negative modes exist in the tensor
sector of perturbations.
Appendix D. Table with longer list of models studied
This appendix provides a table (below) listing the 22 string/M-theory compactifications and the
14 ad hoc low energy models that we have studied most fully, along with the results obtained
(table 1). The methods applied to obtain these results are much like those explained in detail in
the main text and in appendix C for the cases reported there. Models 1–4, 20, 23, 27, and the
d = 4 version of 26 were analyzed in the main text, and models 24, 29, and the d = 4 version
of 28 were discussed in appendix C.
83
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
Table 1. The columns are as follows: model gives the citation where to find the given
supergravity model or indicates the model as described above, fields lists the supergrav-
ity fields taken to be non-trivial, ∂M gives the boundary manifold, E states whether
co-homogeneity one wormholes exist in the model, WD states whether wormholes
ever dominate over the disconnected solution, NM states whether the model has field-
theoretic negative modes and BN states whether the model suffers from brane nucleation
instabilities. In most cases Y/N indicates yes/no. Question marks (?) indicate issues
not investigated or not resolved. In the columns marked by NM and BN the possible
entries are: Y—all wormholes have brane nucleation instabilities/field theoretical neg-
ative modes; YD—all wormholes that dominate over the disconnected solutions have
brane nucleation instabilities/field theoretical negative modes; N—all wormholes are
free of brane nucleation instabilities/field theoretical negative modes; ND—all worm-
holes that dominate over the disconnected solutions are free of brane nucleation instabil-
ities/field theoretical negative modes. Boundary manifolds marked with˜are squashed
at the boundary: e.g. ˜S3 denotes a boundary squashed S3. Entries filled in gray represent
situations for which it does not make sense to fill the respective entry. Model 9 is closely
related to the mass deformation of N = 4 SYM studied in [27].
Model Fields ∂M E WD NM BN
1 [62] φ = χ = 0,Ai(1) = Φδ
i
1, A˜
i
(1) = Φδ
i
1, i = 1, 2, 3 S
3 Y Y ND YD
2 [62] φ = χ = 0,Ai(1) = Φδ
i
1, A˜
i
(1) = Φδ
i
1, i = 1, 2, 3 S˜
3 Y Y ND YD
3 [62] φ 
= 0,χ 
= 0,Ai(1) = A˜i(1) = 0, i = 1, 2, 3 H3 Y ? ? Y
4 [62] φ 
= 0,χ 
= 0,Ai(1) = A˜i(1) = 0, i = 1, 2, 3 H˜3 Y ? ? Y
5 [68] Ai = Φσi,ϕ1 = ϕ2 = −ϕ3, σi = A4 = 0, i = 1, 2, 3 S3 Y Y ND YD
6 [68] Ai = Φσi,ϕ1 = ϕ2 = −ϕ3, σi = A4 = 0, i = 1, 2, 3 S˜3 Y Y ND YD
7 [68] Ai = Φ,ϕ1 = ϕ2 = −ϕ3, σi = A4 = 0, i = 1, 2, 3 T3 Y ? ND Y
8 [68] Ai = A4 = 0,ϕi = ϕ,σi = kxi, i = 1, 2, 3 T
3 N
9 [69, 70] φ,χ H4 Y ? ? Y
10 [71, 72] z4 = −z3 = −z2,β1 = β2 = 0 T4 N
11 [71, 72] z1 = z2 = −z3 = −z4, β1 
= 0, β2 = 0 T4 N
12 [71, 72] z1 = z3, z2 = z4 = β2 = 0, β1 
= 0 T4 N
13 [73] Ai = B0A1,ϕ1 = ϕ2 = 0, i = 1, 2, 3 CP1 ×H2 Y ? ? Y
14 [73] Ai = B0x1 dx2,ϕ1 = ϕ2 = 0, i = 1, 2, 3 T
4 N
15 [73] Ai = B0A1,ϕ1 = ϕ2 = 0, i = 1, 2, 3 CP1 × T2 N
16 [73] Ai = Φσˆi,ϕ1 = ϕ2 = 0, i = 1, 2, 3 S1 × S3 N
17 [73] Ai = Φσˆi,ϕ1 = ϕ2 = 0, i = 1, 2, 3 S1 × S˜3 N
18 [74] Ai = Φiσˆi, i = 1, 2, 3 S3 Y N Y ?
19 [74] Ai = Φ1σˆi,Ai = Φ2σˆ3, i = 1, 2 S3 Y N Y ?
20 IIB φi = Φxi,φ4 = 0, |x| = 1, i = 1, 2, 3 S2 Y Y N Y
21 IIB φ1 + iφ2 = Φ eikx1 ,φ3 + iφ4 = Φ eikx2 T2 Y ? N Y
22 M† φi = Φxi, |x| = 1, i = 1, 2, 3 S2 Y Y N Y
23 EM d = 3,Ai = Φiσˆi, i = 1, 2, 3 S3 Y Y ND
24 EM d = 3, 2AI = LB0εIJK xJ dxK , I = 1, 2, 3 T3 Y Y N
25 ES d = 3, 4, 5,φi = Φxi, |x| = 1, i = 1, . . . , d − 1, μ2 = 0 Sd Y Y ND
26 ES d = 3,φi = Φxi, |x| = 1, i = 1, . . . , 3, μ2 = −2 S3 Y Y ND
27 ES d = 4,φi = Φxi, |x| = 1, i = 1, . . . , 4, μ2 = −3 S4 Y Y ND
28 ES d = 3, 4, 5,φi + iφi+1 = Φ eikxi , i = 1, . . . , d − 1,μ2 = 0 Td Y Y ND
29 ES d = 3,φi + iφi+1 = Φ eikxi , i = 1, 2, 3,μ2 = −2 T3 Y Y ND
30 ES d = 4,φi + iφi+1 = Φ eikxi , i = 1, 2, 3, 4,μ2 = −3 T4 Y Y ND
31 EM d = 4,F1 = F2 S2 × S2 Y ? N
32 E d = 3 S˜3 N
33 EM d = 3,A = Φσˆ3 S˜3 N
34 EM d = 4 CP2 Y ? N
35 EM d = 6 CP3 Y ? N
36 EM d = 3 S1 × S2 N
84
Class. Quantum Grav. 38 (2021) 224002 D Marolf and J E Santos
In most cases, models are described by giving the bibliography reference that describes
them. The consistent truncations leading to the models labeled by IIB were discussed in
section 8. Models labeled by E are pure gravity models, labeled by ES are Einstein-scalar
models and labeled by EM are Einstein–Maxwell models. The model labeled by M† consists
of a reduction of 11-dimensional supergravity on AdS3 × S2 × T6 of the form
ds2 = gab dx
a dxb + L2 dΩ22 + e
√
2φ1 dx21 + e
−√2φ1 dx22
+ e
√
2φ2 dx23 + e
−√2φ2 dx24 + e
√
2φ3 dx25 + e
−√2φ3 dx26 (D.1a)
F(4) =
L
2
d2Ω2 ∧ (dz1 ∧ dz2 + dz3 ∧ dz4 + dz5 ∧ dz6) , (D.1b)
where d2Ω2 is the two-dimensional volume form on the round S2. Perhaps notably, our list does
not include the model described in section 5 of [10] where the disconnected solution remains
to be found in order to determine if the wormhole described there will dominate.
ORCID iDs
Donald Marolf https://orcid.org/0000-0002-4560-4997
Jorge E Santos https://orcid.org/0000-0002-4199-4190
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