Supersymmetric solitons in gauged N 8 supergravity Andr es Anabal on1 Antonio Gallerati2 Simon Ross3 Mario Trigiante24

2025-04-26 0 0 855.66KB 42 页 10玖币

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Supersymmetric solitons in gauged N= 8

supergravity

Andr´es Anabal´on∗1, Antonio Gallerati†2, Simon Ross‡3, Mario Trigiante§2,4

1Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ib´a˜nez, Avenida

Diagonal las Torres 2640, Pe˜nalolen, Santiago, Chile

2Politecnico di Torino, Dipartimento di Scienza Applicata e Tecnologia, corso Duca degli Abruzzi 24,

10129 Torino, Italy

3Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road,

Durham DH1 3LE, U.K.

4Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via Pietro Giuria 1, 10125 Torino, Italy

Abstract

We consider soliton solutions in AdS4with a ﬂat slicing and Wilson loops around

one cycle. We study the phase structure and ﬁnd the ground state and identify

supersymmetric solutions as a function of the Wilson loops. We work in the context

of a scalar ﬁeld truncation of gauged N= 8 supergravity, where all the dilatons

are equal and all the axions vanish in the STU model. In this theory, we construct

new soliton solutions parameterized by two Wilson lines. We ﬁnd that there is a

degeneracy of supersymmetric solutions. We also show that, for alternate boundary

conditions, there exists a non-supersymmetric soliton solution with energy lower

than the supersymmetric one.

∗anabalo@gmail.com

†antonio.gallerati@polito.it

‡simonfross@gmail.com

§mario.trigiante@polito.it

arXiv:2210.06319v2 [hep-th] 7 Feb 2023

Contents

1 Introduction and discussion 3

2 The model 5

2.1 Supersymmetry................................ 6

3 Hairy soliton solutions 9

3.1 Existenceofsolitons ............................. 11

3.2 Relation to earlier solutions . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Constant ﬂux solutions 18

4.1 Quantizationofﬂuxes ............................ 19

5 Supersymmetric solutions 20

5.1 Supersymmetric solutions with ﬁxed ﬂuxes . . . . . . . . . . . . . . . . . 22

5.2 Supersymmetric solutions with ﬁxed charges . . . . . . . . . . . . . . . . 23

6 Phase structure 24

6.1 Euclideanaction ............................... 24

6.2 Fixed ﬂuxes ψ1and ψ2............................ 25

6.3 Fixed qΛ.................................... 26

A Spinor conventions and SUSY for N= 2 models 33

B Analytic solutions for the ﬁxed ﬂuxes 35

C Global properties of the D= 11 background at radial inﬁnity 35

D Remarks on the stability of the hairy soliton vs the AdS soliton at

ﬁxed ﬂuxes 37

References 39

1 Introduction and discussion

An interesting aspect of the AdS/CFT correspondence is the study of the possible

ground states as a function of the boundary conditions. The ﬁrst non-trivial example

is the AdS soliton of [1], which is conjectured to correspond to the ground state of the

theory with a planar boundary, with one direction compactiﬁed on a circle with antiperi-

odic boundary conditions for fermions. In the present paper, we study generalizations

of the AdS soliton when we add Wilson loops for gauge ﬁelds around the circle.

We will work in the context of asymptotically AdS4solutions of a truncation of

N= 8 supergravity. Let us consider a representative of the conformal boundary of a

four-dimensional metric as

ds2

bound. =−dt2+dϕ2+dz2,(1)

where ϕis a periodic coordinate. Fermions can be either periodic or anti-periodic around

the circle parametrized by ϕ. If fermions are anti-periodic, it is possible to construct an

interior solution where the vector ∂ϕhas vanishing norm: this is the AdS soliton. One

might wonder what happens if extra sources are added to this conﬁguration, like, for

instance, a Wilson loop

Φm=IAϕdϕ . (2)

In the following, we will consider the case of anti-periodic boundary conditions on the

ϕcircle, but periodic boundary conditions on the zcircle. As we will see, this allows

supersymmetric solitons for appropriate choices of Wilson loops; it also has the eﬀect

of excluding the possibility of an AdS soliton solution where the zcircle contracts

smoothly in the interior. The relevant solutions will then either have both circles non-

contractible in the interior, or the ϕcircle contractible. The latter solutions, which

generalise the AdS soliton and can be obtained by double analytic continuation from

electrically charged black holes [1–12], are the main focus of our interest.

In the case of the simplest Einstein-Maxwell theory, the conﬁguration has been

known for a while [13,14]. In [12] it was found that some of these conﬁgurations are

supersymmetric and that, at the supersymmetric point, there are two possible solutions,

the soliton and a Poincar´e-AdS solution dressed with a constant Wilson loop. In this

article, we extend this study to the case of gauged N= 8 supergravity, and construct

solutions of its STU model truncation in which the three dilatons are equal and all axions

vanish. In particular, we want to analyse whether the degeneracy of supersymmetric

solutions extends to this more general context.

In the pure Einstein-Maxwell case, supersymmetry was obtained for a single value

of the source (2) [12]. In the model we study here, there are two Wilson lines, with a

one-parameter family of values of the Wilson lines which give supersymmetric solitons.

There are also Poincar´e-AdS solutions and domain wall solutions dressed by constant

Wilson loops, which satisfy the same boundary conditions.

We observe that for the Poincar´e-AdS solutions regularity requires a quantization

condition on the Wilson loops which was missed in [12]. This arises from considering

the uplift of the conﬁguration to a M4×S7solution of M theory, where the Wilson

loops parametrise a shift on the S7as we go around the ϕcircle.

For ﬁxed ﬂux boundary conditions, the coexistence of the soliton and the Poincar´e-

AdS and domain wall solutions leads to a degeneracy of supersymmetric solutions as

in [12]. We also consider the alternate boundary condition of ﬁxed currents on the

boundary – we will in the sequel refer to these boundary conditions as ﬁxed charge –

although this is actually correct only for the Euclidean continuation, as the currents

are spacelike in the Lorentzian solution. We ﬁnd that, for supersymmetry-preserving

ﬁxed charge boundary conditions, there are two distinct soliton solutions, leading to

a new kind of degeneracy of supersymmetric solutions (the Poincar´e-AdS and domain

wall solutions do not satisfy these boundary conditions, so there is no degeneracy at

ﬁxed charge in the previous case studied in [12]).

Finally, we ﬁnd that the non-supersymmetric solutions of [12] also satisfy the bound-

ary conditions which give supersymmetric solutions for ﬁxed charge, and one branch of

those solutions has lower energy than the supersymmetric solutions. This is surprising

as we would expect the supersymmetric solutions to saturate a BPS bound, forbidding

the existence of solutions with lower energy. The BPS bound for these alternate bound-

ary conditions has however not been explicitly worked out as far as we are aware. We

discuss this result in light of the positive energy theorem [15,16], which implies that the

energy of a supersymmetry preserving solution is lower than the energy of any other

solution satisfying the same boundary conditions. We show that our result is not in

contradiction with this general property. The central point of the argument is that a

necessary condition for the positive energy theorem to apply is the existence, for the

non-supersymmetric solution, of an asymptotic Killing spinor which coincides, up to

O(1/r2) terms at radial inﬁnity, with the Killing spinor of the supersymmetric one.

Since the latter has antiperiodic boundary conditions along the circle at inﬁnity, in or-

der for the positive energy theorem to apply, the non-supersymmetric solutions should

admit an asymptotic Killing spinor with the same property at the boundary. As we

shall prove, this is the case only if the charges at inﬁnity have speciﬁc values, for which

the energy of the non-supersymmetric solution exceeds that of the supersymmetric one.

In summary, there is no contradiction with the positive energy theorem if we include

among the boundary conditions those applying to the asymptotic Killing spinor.

Another important direction for future work is to ﬁnd a more general understanding

of the degeneracy of the susy solutions. We believe this is a generic feature of such

boundary conditions; we intend to provide a general proof in a forthcoming paper.

It is also possible to construct black holes in this theory [17–23] and endow them

with Wilson lines along the lines of [24]. This will provide an even more complete phase

diagram of this model that we leave to analyze in the future.

The outline of the paper is as follows. In the next Section 2, we review the model

under consideration. In Section 3, we present the soliton solutions within this model,

and explain their relation to the solutions of [12]. In Section 4, we discuss the solutions

with constant ﬂuxes, and explain the quantisation of the ﬂux from demanding a well-

behaved action on the S7factor in the uplift. In Section 5, we ﬁnd the supersymmetric

solutions for both ﬁxed ﬂux and ﬁxed charge boundary conditions. In Section 6, we

describe the phase structure for the diﬀerent boundary conditions, and point out that for

supersymmetric ﬁxed charge boundary conditions there are both supersymmetric and

non-supersymmetric solutions, with a surprising family of non-supersymmetric solutions

of lower energy and free energy than the supersymmetric ones. The consistency of this

result with the positive energy theorem for asymptotically AdS solutions is discussed in

subsection 6.3.1.

2 The model

We are interested in studying the dilatonic sector of the STU model of the SO(8)-gauged,

N= 8 supergravity with action:

S=1

2κZd4x√−g R−

i=1

(∂Φi)2

2+2

L2cosh (Φi)−1

i=1

X−2

i¯

i!,(3)

where ¯

Fiare two forms, related with gauge ﬁelds in the standard way

Fi=d¯

Ai, Xi=e−1

2~ai·~

Φ,~

Φ = (Φ1,Φ2,Φ3),(4)

and

~a1= (1,1,1) , ~a2= (1,−1,−1) , ~a3= (−1,1,−1) , ~a4= (−1,−1,1) .(5)

We will be interested in purely magnetic solutions, in which case it is consistent to trun-

cate the axions to zero. The Lagrangian (3) can be obtained from the compactiﬁcation

of eleven dimensional supergravity over the seven sphere with the ansatz [25]

ds2

11 =˜

∆2/3ds2

4+ 4 L2˜

∆−1/3

i=1

X−1

i dµ2

i+µ2

idϕi+1

2L¯

Ai2!,(6)

F=−1

L4

i=1 X2

iµ2

i−˜

∆Xi+L X−1

i?4dXi∧dµ2

i−

−4L2X

X−2

iµidµi∧dϕi+1

2L¯

Ai∧?4¯

Fi,(7)

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摘要：

SupersymmetricsolitonsingaugedN=8supergravityAndresAnabalon*1,AntonioGallerati2,SimonRoss3,MarioTrigiante§2,41DepartamentodeCiencias,FacultaddeArtesLiberales,UniversidadAdolfoIba~nez,AvenidaDiagonallasTorres2640,Pe~nalolen,Santiago,Chile2PolitecnicodiTorino,DipartimentodiScienzaApplicataeTecnol...

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Supersymmetric solitons in gauged N 8 supergravity Andr es Anabal on1 Antonio Gallerati2 Simon Ross3 Mario Trigiante24

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