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Supersymmetric Wormholes in String Theory
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Supersymmetric Wormholes in String Theory
Davide Astesiano and Fri ?rik Freyr Gautason
Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavík, Iceland
(Received 22 September 2023; revised 19 February 2024; accepted 21 March 2024; published 15 April 2024)
We construct a large family of Euclidean supersymmetric wormhole solutions of type IIB supergravity
which are asymptotically AdS 5×S5. The solutions are constructed using consistent truncation to
maximally gauged supergravity in five dimensions which is further truncated to a four scalar model.
Within this model we perform a full analytic classification of supersymmetric domain wall solutions with
flat Euclidean domain wall slices. On each side of the wormhole, the solution asymptotes to AdS 5dual to
N?4supersymmetric Yang-Mills deformed by a supersymmetric mass term.
DOI: 10.1103/PhysRevLett.132.161601
Introduction. —The Euclidean path integral for quantum
gravity is an important topic of research and for low-
dimensional theories such as JT gravity, has recently lead tomany fruitful results; see, for instance, Ref. [1]. It has
become clear that in low-dimensional theories it makes
sense to sum over saddle points with different topologies
[2]. Still, in higher dimensions for standard Einstein-Hilbert
gravity (coupled to matter) the rules remains somewhat
unclear. The story in higher dimensional gravity theories
can be different from low-dimensional ones without lead-ing to obvious inconsistencies.
In this regard, the role of wormholes, as possible saddle
points of the path integral is still an important open problem
[3]. The processes that involve wormholes pose puzzles for
unitarity of the quantum system and nonfactorization ofcorrelation functions in the holographic dual [4]. Moreover,
the existence of wormholes indicates that probability
amplitudes to produce or absorb baby universes are non-trivial which may lead to issues for the Swamplandprogram [5,6] .
To improve our understanding it is necessary to provide
the embedding of higher dimensional Euclidean worm-
holes in string theory and AdS/CFT. In this way, various
ideas regarding the semiclassical formulation of gravity canbe put to a test. Research in this direction was initiated in
some earlier works [7,8] .
In order to construct Euclidean wormhole geometries we
generally need a source of negative Euclidean energy. In
string theory there is a natural way to obtain the requirednegative energy, which is to consider axion fields [9]. When
a Lorentzian theory containing axions is analyticallycontinued to Euclidean, the axion kinetic term may become
negative definite which gives rise to the required negativeenergy-momentum tensor, see also [10–12].
In the present work [13] we will consider a consistent
truncation of type IIB supergravity on S
5down to five-
dimensional maximal supergravity coupled to SO(6) gaugegroup [16–18]. The consistent truncation means that every
solution of the five-dimensional model can be “uplifted ”to
a solution of full type IIB supergravity [15,19,20] .O u r
model will be a further (consistent) truncation to a fourscalar theory coupled to AdS gravity originally introduced
in[21,22] to study the holographic duals to N?4
Supersymmetric Yang-Mills (SYM) deformed by a massparameter. As we will see, the model gives rise to singulardomain walls, as well as regular Euclidean wormholes that
have much in common with the original axionic wormholes
of[9].
The embedding of axionic wormholes as AdS compac-
tifications of 10D (or 11D) supergravity has been discussed
recently in the literature [23–27]. In short, the existence of
the wormhole solution relies on the existence of moduliscalars, whose metric is not necessarily positive definite.When we Wick rotate to Euclidean signature in space-time,the axions get flipped signs [28], while the other scalar
fields remains untouched [12,29] .
In the model studied in this paper, we will encounter a
similar feature when rotating to Euclidean signature. Onlyone of the four scalars is a modulus that happens to be thefive-dimensional dilaton. The dilaton is related to the Yang-Mills coupling constant in the dual N?4SYM theory.
The other scalars in our model have a nontrivial potentialand play a crucial role in our construction of wormholes.Their presence makes the equations of motion much more
complicated than for standard axionic wormholes.
However with the help of first order BPS (Bogomol'nyi,Prasad, Sommerfield) equations that ensure supersymmetryof the solutions, we will be able to find supersymmetricEuclidean wormholes. In more detail, the solution we
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI. Funded by SCOAP3.PHYSICAL REVIEW LETTERS 132, 161601 (2024)
0031-9007 =24=132(16) =161601(6) 161601-1 Published by the American Physical Societyconstruct is a five-dimensional domain wall of the form
ds2
5?dr2?e2A?r?ds2
R4; ?1?
where d s2
R4is the flat metric on R4and the metric function
Aonly depends on r. The R4can be compactified to T4
when considering the Euclidean gravity path integral. The
solution we find will be described by a metric function A?r?
that approaches the standard asymptotic AdS form A?r??
/C6r=2Lfor both r→/C6∞connected by a region of smooth
region of non-AdS space. Close to the asymptotic boundary
we find a pair of (Euclidean) AdS 5spaces dual to N?4
SYM. The scalar fields and the metric have asymptoticform which is consistent with masses for the three chiral
fields in N?4being turned on. Supersymmetry is there-
fore broken from N?4toN?1. The two QFTs dual to
each side of the wormhole are both N?1
/C3, but with
different Yang-Mills coupling constants and different vevsturned on. A special line of solutions exists where the
configuration is slightly more symmetric and the two
boundary theories have the same vevs. On this line there
is a very special point where the neck of the wormhole
shrinks to zero size and the metric becomes singular and we
recover the well-known GPPZ (Girardello, Petrini, Porrati,
Zaffaroni) solution [30,31] .
An important question when faced with wormhole
solutions such as these ones is whether they dominate
over the corresponding “disconnected geometry. ”Since we
have been focusing on supersymmetric solutions, the
disconnected geometry should also preserve supersym-
metry. In our analysis we have been able to fully classify
solutions to the BPS equations subject to the metric ansatz
(1). It turns out that for a given set of boundary conditions
which allow for a wormhole solution, there is no corre-
sponding disconnected solution. Disconnected solutions
could perhaps be found by relaxing some of the isometries
built into the metric ansatz (1), but we have not carried out a
general analysis. It is straightforward to check that for our
BPS solutions, the regularized on-shell Euclidean action
vanishes.
5D supergravity. —The supergravity model considered
here is a four scalar truncation of maximal 5D supergravity
with SO(6) gauge group [16–18]. The 5D SO(6) gauged
maximal supergravity has been shown to arise as consistent
truncation of type IIB supergravity on S
5[15,19,20] and so
any solution of the maximal supergravity can be embedded
into type IIB supergravity.
The four scalar truncation discussed presently was first
introduced in the holographic study of the N?1mass
deformation of N?4SYM with all three mass parameters
taken to be equal [21,22,32] . When all masses are equal the
QFT possesses SO(3) flavor symmetry which (if we
assume it is not spontaneously broken) can be utilized
on the supergravity side to truncate the maximal theory
such that the bosonic sector contains a metric and eightscalar fields [21]. A further discrete symmetry can be
imposed to truncate the theory even further leaving onlyfour scalar fields apart from the metric.
As the name suggests, the scalars of the model
parametrize a four-dimensional subspace of the full 42dimensional scalar manifold E
6?6?=USp?8?. This subspace
consists of two copies of the Poincar? e disc which we
parametrize with two complex scalar fields z1;2[33].
The five-dimensional supergravity action of the four
scalar model takes the form
S?1
16πGNZ
?/C0
R?2Ki?|?μzi?μ?z?|?P/C1
; ?2?
where the scalar potential is
P?1
2eK/C18
Ki?|DiWD?|?W?8
3jWj2/C19
; ?3?
with the K?hler covariant derivative defined as Dif?
??i??iK?f, and the K?hler metric defined by Ki?|??i??|K
and its inverse is Ki?|. We have written the theory in terms of
the K?hler potential Kand a holomorphic superpotential W
which are given by
K??X2
i?1log?2Imzi?;W?3gz2?z1?z2?:?4?
The theory exhibits the scaling symmetry zi?λziwhich
leaves the action invariant. This is nothing but the dilatonicshift symmetry.
The maximally supersymmetric vacuum solution of the
maximal five-dimensional supergravity is obtained as acritical point of this model by setting z
1?z2?ieφwhere
φis the constant value given to the five-dimensional
dilaton. For the vacuum solution, the scalar potential takes
the value P??3g2and therefore the metric is AdS 5with
length scale L??2=g?.
BPS equations and wormhole solutions. —We are inter-
ested in finding flat sliced supersymmetric domain wall
solutions to the equations of motion of the four scalarmodel. To this end we assume that the five-dimensionalmetric takes the form (1)and assume that all scalar fields as
well as the metric function Aare only functions of the radial
variable r. A supersymmetric solution must satisfy the first
order equations [32]
E
A≡A0?1
3W?0;Ei≡?zi?0?Ki?|??|W?0;?5?
where the real superpotential Wis defined as W?
eK=2jWj. We have verified that all solutions to the BPS
equations are also solutions to the five-dimensional equa-tions of motion. It should be noted at this point that inLorentzian supergravity ?z
?{is the complex conjugate of ziPHYSICAL REVIEW LETTERS 132, 161601 (2024)
161601-2and the same holds true for Wand ?W. In this Letter we will
also consider Euclidean solutions where ?z?{is best treated as
independent from zi. In general ziand ?z?{still represent 2
real degrees of freedom in total. This feature has been
discussed previously in, e.g., [22] but will become more
apparent later when we discuss the explicit solution to theBPS equations.
In order to simplify the system of BPS equations,
we introduce new field variables z
1?ieφ?3α?iθ1and
z2?ieφ?α?iθ2. The new scalar fields have a clear inter-
pretation from the perspective of the holographic dual field
theory. In particular, φis the five-dimensional dilaton and is
dual to the marginal Yang-Mills coupling, αis dual to a
scalar bilinear operator transforming in the 200representa-
tion of SO(6) and θ1;2are dual to two fermion bilinear
operators transforming in the 10?10representation.
It turns out to be useful to further define new sets of
variables t1;2?tanθ1;2in order to eliminate most of the
trigonometric functions. Even with these new variables the
BPS equations are quite lengthy and difficult to analyze. Inorder to make progress we replace the field αby a new
variable Xdefined by
X≡1
2?1?t2
2?/C18
1?t1t2???????????? ?
1?t2
1q ??????????? ?
1?t2
2q
cosh 4α/C19
:?6?
This definition of Xmay seem ad hoc at first but it is
closely related to the real superpotential W. With these
definitions the BPS equations take the form
4????
Xp
?t0
1??3g/C0
t2?t1?2Xt1?1?t2
2?/C1
;
4????
Xp
?t0
2??g/C0
t1?t2?6Xt2?1?t2
2?/C1
;
4????
Xp
?X0??8gX?X?1?;
4????
Xp
?A0???2gX?1?t2
2?: ?7?
Writing the BPS equations in these coordinates has
simplified them significantly enabling us to fully solve
them. Note that the dilaton has been decoupled completelyfrom the system as it does not appear on the right-hand side
of any of the equations. This does not mean that the dilaton
is constant, however, as its BPS equation has a complicatedright-hand side.
Recall now that the scaling symmetry present in our
model implies the existence of a constant of motion [34]
j??
g3
64e3A?t1?3t2?; ?8?which implies that we do not have to solve explicitly the
equations for both t1andt2, only one combination of them
suffices. For this purpose we identify another combinationof the tscalars and solve for Xin terms of the new variable
Y≡g3
64e3A?t1?t2?;X ?Y2
k?Y2; ?9?
where kis a real integration constant. We can rescale the
metric function such that without loss of generality we can
consider three distinct values k?f?1;0;1g. Next we use
(9)to write
dA
dY?1
2Y
k?Y2/C0
1?256g?6e?6A?j?Y?2/C1
:?10?
Finally, we remark that in the Ycoordinate, the five-
dimensional metric takes the form
ds2
5?dY2
g2?Y2?k??e2A?Y?ds2
R4: ?11?
A wormhole solution is obtained if the metric function
e6Ahas two AdS 5asymptotic regions (for jYjlarge) and is
otherwise positive. This only happens if k?1and the
metric function takes the form
g6
26e6A?4/C0
2jY3?3Y2?j2?2?2a?Y2?1?3=2/C1
:?12?
Since e6Ahas at most two real roots, we have to ensure that
the discriminant of the polynomial ?2jY3?3Y2?j2?2?2?
4a2?Y2?1?3is negative implying it has no real roots.
Combined with the condition that a>jjjwe find wormhole
solutions if and only if 1?j2=2that the scalar field αis imaginary when the above condition
is satisfied. In fact the condition for αbeing real is that
1?j2≥a2. The boundary of which is where the scalar α
vanishes throughout and can be identified with the GPPZ
solutions [30]. The two regions are completely nonoverlap-
ping. It is interesting to note that the GPPZ solution withj?0anda?1(orλ?1in the notation of [32])i s
infinitesimally close to being a wormhole and can be viewed
as the limiting solution where the wormhole neck shrinks to
zero size.
In addition to the αscalar being imaginary, the dilaton is
also imaginary. In order to see that we have to integrate theBPS equation for the dilaton which for k?1takes the form
φ?Y??φ
0?ZY
?∞?3i??????????????????????
a2?j2?1p
y?y?j?
2?????????????
y2?1p
?a?????????????
y2?1p
?jy?1?h
y2?a?????????????
y2?1p
?3??a?????????????
y2?1p
?j?y2?3?y?1idy: ?13?PHYSICAL REVIEW LETTERS 132, 161601 (2024)
161601-3We have not been able to perform the integral analyti-
cally but it is easy to do numerically. We display a plot of a
sample solution in Fig. 1. A generic feature of the worm-
hole solutions is that they are not symmetric around Y?0,
even for j?0. This is clearly observed from Fig. 1where
the dilaton is far from being symmetric around Y?0.
Field theory interpretation and on-shell action. —Recall
that the scalar fields α,φ, and θ1;2have a direct relation to
operators in the dual field theory. More precisely if θ1?
?3?1??2andθ2??1??2, and we think of N?4
SYM in N?1language, then ?2is dual to the gaugino
bilinear and ?1is dual to the three chiral multiplet fermion
bilinear. The latter are all equal since we assumed thatthe SO(3) flavor symmetry preserved by the equal mass
N?1
/C3Lagrangian is not spontaneously broken. By
performing a expansion of the fields around the AdS 5
asymptotics we can identify the sources and vevs given tothe dual operators for our solution. A general UVexpansioncompatible with the BPS equations takes the form
A??
1
2log??O??2?; ?1?m?1=2?O??2?;
?2?w?3=2?O??2?; α?v??O??2?; ?14?
andφis constant at leading order. Here ?is the small
parameter controlling the distance from the asymptotic
boundary. From this expansion we see that the parameter m
is the mass given to the chiral fields whereas wis the
gaugino vev and vis the so-called chiral condensate, i.e.,
the vev of the scalar bilinear in the chiral multiplets.
For the wormhole solutions with k?1,w eh a v et w o
asymptotic regions that are located at Y→/C6∞. Expanding
our solution we find the dimensionless quantities
w/C6
2m3??j2?1?/C6aj;v2
m4??j2?1??a2;?15?where we have denoted the two gaugino condensates that
are encountered in the two asymptotic regions Y→/C6∞by
w/C6. The dimensionless chiral condensate v=m2,i st h e
same in both regions. Regular wormhole solution exist
only when the chiral condensate v=m2is imaginary. The
conclusion is that the Euclidean wormhole solution is abulk geometry that connects two copies of mass-deformed
N?4SYM where some of the boundary conditions
(including the Yang-Mills coupling constant) are different.
An important aspect for the evaluation of the gravita-
tional partition function, and of the free energy of the dualtheory, is the on-shell action of our wormholes solutions.Adding the Gibbons-Hawking term to the action andperforming a partial integration, the action can be rewrittenin terms of the squared BPS equations
L?L
GH??e4A/C2
12E2
A?2Ki?|Ei?E?|/C3
?2?r?e4AW?;
where Wis the real superpotential we defined before
W?eK=2jWj. Evaluating this on-shell, the BPS equations
set to zero the first two terms, leaving only the total
derivative. The total derivative term leads to a divergent
expression which must be regulated. As explained in[22,36,37] the correct supersymmetric counterterm (when
the holographic boundary is flat) should be chosen toexactly cancel the total derivative. This implies that for allsupersymmetric regular wormhole solutions found in thisLetter, the on-shell action vanishes.
Final comments. —As anticipated in the introduction, our
wormhole solutions are supported by a negative term in the
energy-momentum tensor. In fact, the dilaton φand the
field αare imaginary while preserving the reality of the
metric as well as the action. This is interpreted as a Wickrotation on target space which must be simultaneouslyperformed when Wick rotating space-time. The target spacemetric exhibits a pair of translation symmetries for αandφ,
which therefore appear as axions. However it must be noted
that the scalar potential depends nontrivially on αand so the
shift “symmetry ”ofαis not a true symmetry of the theory.
Nevertheless, according to the prescription in [24,26] , the
Wick rotation of Lorentzian supergravity to Euclideanshould be accompanied with a similar Wick rotation intarget space α→iαandφ→iφ[38]. The Wick rotation
affects the potential but is still real.
While the space-time signature becomes Euclidean, the
target space signature is now Lorentzian. This is not particu-
larly surprising if we remember that in Euclidean signature,
the R symmetry of N?4SYM is not SO(6) but rather
SO(1,5). This is also consistent with the fact that the holo-
graphic dual to Euclidean N?4SYM is described by so-
called type IIB
/C3supergravity [39–41].T h i st h e o r yh a sa
metric with space-time signature (9,1) but some of the form
fields have negative kinetic terms (and can therefore be–2 –1 0 1 2–1012
Y
FIG. 1. A plot of a wormhole solution for a?2.6andj?0.8.
The function g2e2A=10is drawn in gray, cosh 4αis drawn in solid
black, and cosh 4φin dashed black.PHYSICAL REVIEW LETTERS 132, 161601 (2024)
161601-4thought of as being analytically continued from standard type
IIB supergravity). In particular the ten-dimensional axion-
dilaton parametrize the coset space SU ?1;1?=SO?1;1?.I ti sa
subject of future work to uplift the wormholes to ten
dimensions using the results of [15,19,20] . The fact that
the five-dimensional dilaton is imaginary may appear worri-some when interpreted in ten dimensions. Even though the
relation between the five-dimensional dilaton and the ten-
dimensional one is rather complicated, asymptotically they
are simply related. It would therefore appear that the ten-
dimensional dilaton is imaginary for the wormhole solutions
which is troubling. It turns out there is a simple remedy for this
problem by employing an SL ?2;R?transformation that
renders the dilaton real but turns on an imaginary axion
consistent with being a solution of type IIB
/C3[35].
Since our solutions preserve supersymmetry and are
regular we do not expect any instabilities to arise and
question the validity of them. By all accounts they should
then contribute to the Euclidean path integral. Since we did
not find disconnected geometries with the same boundaryconditions as the wormholes, we are unable to answer
whether the wormholes dominate or not. If they dominate,
then it leads to the well-known factorization puzzle inholography [7,8] in this case for deformations of AdS
5dual
to theN?1/C3theory. How this puzzle is resolved is a open
question at this stage. One possibility is that fermion zero
modes in the spectrum cause the wormhole contribution tothe path integral to vanish. This was indeed observed in
[42] and in that case it was related to supersymmetry being
broken.
We are grateful to Nikolay Bobev, Valentina G. M.
Puletti, Krzysztof Pilch, Thomas Van Riet, Watse
Sybesma, and Lárus Thorlacius for useful discussions.We thank Nikolay Bobev, Thomas Van Riet, and Lárus
Thorlacius for comments on the manuscript. F. F. G. and
D. A. are supported by the Icelandic Research Fund,
Rannís, under Grant No. 228952-052. F. F. G. is partially
supported by grants from the University of IcelandResearch Fund.
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[34] Identifying this constant of motion is quite tricky because
naively the Noether charge seems to vanish identically.
However by slightly deforming the metric ansatz and
considering curved domain walls the correct conserved
quantity (8)can be derived [35].
[35] D. Astesiano and F. F. Gautason Supersymmetric worm-
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Phys. 05 (2023) 032.PHYSICAL REVIEW LETTERS 132, 161601 (2024)
161601-6
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