susy q-balls and boson stars in anti-de sitter space-time

AdS/CFT correspondenceSUSY Q-balls in AdS background

SUSY boson stars in AdS backgroundConclusion

SUSY Q-Balls and Boson Stars in Anti-deSitter space-time

Jürgen Riedelin Collaboration with Betti Hartmann, Jacobs University Bremen

School of Engineering and ScienceJacobs University Bremen, Germany

DPG TALK 2012Göttingen, March 1st 2012

Jürgen Riedel & Betti Hartmann SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time



Outline

1 AdS/CFT correspondence

2 SUSY Q-balls in AdS background

3 SUSY boson stars in AdS background

4 Conclusion




AdS/CFT correspondence

Important result from StringTheory (Maldacena, 1997):A theory of classical gravity in (d + 1)-dimensionalasymptotically Anti-de Sitter (AdS) space-time is dual to astrongly-coupled, scale-invariant theory (CFT) living onthe d-dimensional boundary of AdSAn important example: Type IIB string theory in AdS5× S5dual to 4-dimensional N = 4 supersymmetric Yang-MillstheoryOne can use classical gravity theory, i.e. weakly-coupled,to study strongly coupled quantum field theories




Holographic conductor/ superconductor

Taken from arxiv: 0808.1115

Boundary of SAdS ≡ AdS

Dual theory“lives” here

r → ∞

r

x,yr=r

h horizon

Temperature represented bya black hole

Chemical potentialrepresented by a chargedblack hole

Condensate represented bya non-trivial field outside theblack hole horizon if T < Tc

⇒ One needs an electricallycharged plane-symmetrichairy black hole




The model

Action ansatz:S =

∫dx4√−g

(R + 6

`2 − 14 FµνFµν − |DµΦ|2 −m2|Φ2|

)Metric with r = rh event horizon (AdS for r →∞) +negative cosmological constant Λ = −3/`2

ds2 = −g(r)f (r)dt2 +dr2

f (r)+ r2(dx2 + dy2)

Ansatz: Φ = Φ(r), At = At (r)

Presence of the U(1) gauge symmetry allows to gaugeaway the phase of the scalar field and make it real




Holographic insulator/ superconductor

double Wick rotation (t → iχ, x → it) of SAdS with rh → r0

ds2 = dr2

f (r) + f (r)dχ2 + r2(−dt2 + dy2

)with f (r) = r2

`2

(1− r3

0r3

)It is important that χ is periodic with period τχ = 4π`2

3r0

Scalar field in the background of such a soliton has astrictly positive and discrete spectrum (Witten, 1998)

There exists an energy gap which allows theinterpretation of this soliton as the gravity dual of aninsulatorAdding a chemical potential µ to the model reduces theenergy gap




Outline




4 Conclusion




The e = 0 limit

In the case of vanishing gauge coupling constant e:

The scalar field decouples from gauge fieldOne cannot use gauge to make scalar field realThe simplest ansatz for complex scalar field:φ(r) = φeiωt

This leads to Q-balls and boson stars solutions




The model for G = 0

SUSY potential U(|Φ|) = m2η2susy

(1− exp

(−|Φ|2/η2

susy))

Metric ds2 = −N(r)dt2 + 1N(r)dr2 + r2

(dθ2 + sin2 θdϕ2

)with N(r) = 1 + r2

`2and ` =

√−3/Λ

Using Φ(t , r) = eiωtφ(r), rescaling

Equation of motion φ′′ = −2r φ′ − N′

N φ′ − ω2

N2φ+ φ exp(−φ2)N

Power law for symptotic fall-off for Λ < 0:

φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2

Charge and mass Q = 8π∫∞

0 φr2dr andM = 4π

∫∞0

[ω2φ2 + φ′2 + U(φ)

]r2dr




First results of the numerical analysis

ω

M

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

5010

020

050

010

0020

00

Mass over Omega

Λ= 0= −0.01= −0.02= −0.025

ωM

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

5010

020

050

010

0020

00

Charge over Omega

Λ= 0= −0.01= −0.02= −0.025

Figure : Properties of SUSY Q-balls in AdS background mass M (left) and charge Q(right) versus frequency ω for various values of Λ





φ(0)

M

0 2 4 6 8 10

110

100

1000

1000

0

Mass over Phi(0)

Λ= 0= −0.01= −0.02= −0.025

φ(0)Q

0 2 4 6 8 10

110

100

1000

1000

0

Charge over Phi(0)

Λ= 0= −0.5= −0.−1= −5

Figure : Properties of SUSY Q-balls in AdS background mass M (left) and charge Q(right) versus scalar field function at the origin φ(0) for various values of Λ





M

Q

200 500 1000 2000 5000 10000 20000200

500

2000

5000

2000

050

000

Charge over Mass

Λ= 0= −0.01= −0.02= −0.025

ω

φ(0)

0.2 0.4 0.6 0.8 1.0 1.2

02

46

810

Phi(0) over Omega

Λ= 0= −0.01= −0.02= −0.025

Figure : Properties of SUSY Q-balls in AdS background mass M versus charge Q(left) and the scalar field function at the origin φ(0) versus frequency ω (right) forvarious values of Λ





M

Con

dens

ate

0 5000 10000 15000

0.01

00.

015

0.02

00.

025

Condensate over Mass

Λ= −0.03= −0.04= −0.05= −0.075

QC

onde

nsat

e0 5000 10000 15000 20000

0.01

00.

015

0.02

00.

025

Condensate over Charge

Λ= −0.03= −0.04= −0.05= −0.075

Figure : Condensate O1∆ over Mass M (left) and charge Q (right) for various values

of Λ





φ(0)

Con

dens

ate

0 2 4 6 8 10

0.01

00.

015

0.02

00.

025

Condensate over Phi(0)

Λ= −0.03= −0.04= −0.05= −0.075

Figure : Condensate O1∆ as function of the scalar field at φ(0) for various values of Λ




Outline




4 Conclusion




SUSY potential U(|Φ|) = m2η2susy

(1− exp

(−|Φ|2/η2

susy))

The coupling constant κ is given with κ = 8πGη2susy

Metricds2 = −A2(r)N(r)dt2 + 1

N(r)dr2 + r2 (dθ2 + sin2θdϕ2) with

N(r) = 1− 2n(r)r − Λ

3 r2 and ` =√−3/Λ

Using Φ(t , r) = eiωtφ(r) and rescalingEquations of motionn′ = κ

2 r2(

N(φ′)2 + ω2φ2

A2N + 1− exp(−φ2))

,

A′ = κr(ω2φ2

AN2 + Aφ′)

and(r2ANφ′

)′= −ω2r2

AN + r2Aφexp(−φ2)


φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2




Calculating the mass


φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2

The mass in the limit r � 1 and κ > 0 isn(r � 1) = M + n1φ

2∆r2∆+3 + ... with n1 = −Λ∆2+3

6(2∆+3)

For the case κ = 0 the Mass M is with n(r) ≡ 0, A(r) ≡ 1:M =

∫d3xT00 = 4π

∫∞0

[ω2φ2 + N2(φ′)2 + NU(φ)

]r2dr

The charge Q is given for all values of κ as:Q = 8π

∫∞0

ωr2

AN dr





ω

M

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Mass over Omega

κ= 0.0= 0.001= 0.01= 0.05= 0.1

ωQ

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Charge over Omega

κ= 0.0= 0.001= 0.01= 0.05= 0.1

Figure : Properties of SUSY boson stars in AdS background mass M (left) andcharge Q (right) versus frequency ω for various values of κ and fixed Λ = 0.0





φ(0)

Q

0 2 4 6 8 10

1050

500

5000

Charge over Phi(0)

κ= 0.0= 0.001= 0.01= 0.05= 0.1

ωφ(

0)

0.2 0.4 0.6 0.8 1.0

05

1015

Phi(0) over Omega

κ= 0.0= 0.001= 0.01= 0.05= 0.1

Figure : Properties of SUSY boson stars in AdS background charge Q versus φ(0)

(left) and φ(0) versus frequency ω (right) for various values of κ and fixed Λ = 0.0





ω

Q

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Charge over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

ωQ

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Charge over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

Figure : Properties of SUSY boson stars in AdS background charge Q versusfrequency ω for various values of κ and fixed Λ = −0.001 (left) and fixed Λ = −0.01(right)





ω

φ(0)

0.2 0.4 0.6 0.8 1.0

05

1015

20

Phi(0) over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

ωφ(

0)

0.2 0.4 0.6 0.8 1.0

05

1015

20

Phi(0) over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

Figure : Properties of SUSY boson star in AdS background φ(0) versus frequency ωfor various values of κ and fixed Λ = −0.001 (left) and fixed Λ = −0.01 (right)





ω

Q

0.2 0.4 0.6 0.8 1.0 1.2 1.4

1050

500

5000

Charge over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

ωQ

0.2 0.4 0.6 0.8 1.0 1.2 1.4

1050

500

5000

Charge over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

Figure : Properties of SUSY boson stars in AdS background charge Q versusfrequency ω for various values of Λ and fixed κ = 0.0 (left) and fixed κ = 0.01 (right)





ω

φ(0)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

02

46

810

Phi(0) over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

ωφ(

0)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

02

46

810

Phi(0) over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

Figure : Properties of SUSY boson star in AdS background φ(0) versus frequency ωfor various values of Λ and fixed κ = 0.0 (left) and fixed κ = 0.01 (right)




Outline




4 Conclusion




Summary of first Results

Shift of ωmax for Q-balls and boson stars to higher valuesfor increasingly negative values of Λ, i.e.ωmax →∞ for Λ→ −∞The minimum value of the frequency for Q-balls isωmin = 0 for all Λ

The minimum value of the frequency for boson starsωmin increases for increasingly negative values of Λ

The curves mass M over frequency ω and charge Qversus ω for Q-balls and boson stars show

M → 0 for ω → ωmaxQ → 0 for ω → ωmax




Summary of first Results continued

For boson stars the cosmological constant Λ ’kills’ thelocal maximum of the charge Q and Mass M near ωmax ,similarly as large values of κ

The curve of the condensate for Q-balls, i.e. O1∆ as a

function of the scalar field φ(0), has qualitatively thesame shape as in Horowitz and Way, JHEP 1011:011, 2010[arXiv:1007.3714v2]




Outlook

Studying the condensate of boson stars in AdS withSUSY potentialInterpreting the condensate in the context of CFTStudying Q-balls and boson stars in AdS in (d+1)dimensionsStudying rotating boson stars in AdS with SUSYpotential


susy q-balls and boson stars in anti-de sitter space-time

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