Conclusion
Outline
1 AdS/CFT correspondence
2 SUSY Q-balls in AdS background
3 SUSY boson stars in AdS background
4 Conclusion

Conclusion
Important result from StringTheory (Maldacena, 1997):
A theory of classical gravity in (d + 1)-dimensional
asymptotically Anti-de Sitter (AdS) space-time is dual to a
strongly-coupled, scale-invariant theory (CFT) living on
the d-dimensional boundary of AdS
An important example: Type IIB string theory in AdS5 × S5
dual to 4-dimensional N = 4 supersymmetric Yang-Mills
theory
One can use classical gravity theory, i.e. weakly-coupled,
to study strongly coupled quantum ﬁeld theories

Conclusion
Holographic conductor/ superconductor
Taken from arxiv: 0808.1115
Boundary
of SAdS
≡
AdS
Dual theory
“lives” here
r → ∞
r
x,y
r=rh
horizon
Temperature represented by
a black hole
Chemical potential
represented by a charged
black hole
Condensate represented by
a non-trivial ﬁeld outside the
black hole horizon if T < Tc
⇒ One needs an electrically
charged plane-symmetric
hairy black hole

Conclusion
The model
Action ansatz:
S = dx4√
−g R + 6
2 − 1
4 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 gauge
away the phase of the scalar ﬁeld and make it real

Conclusion
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
0
r3
It is important that χ is periodic with period τχ = 4π 2
3r0
Scalar ﬁeld in the background of such a soliton has a
strictly positive and discrete spectrum (Witten, 1998)
There exists an energy gap which allows the
interpretation of this soliton as the gravity dual of an
insulator
Adding a chemical potential µ to the model reduces the
energy gap

Conclusion
The e = 0 limit
In the case of vanishing gauge coupling constant e:
The scalar field decouples from gauge field
One cannot use gauge to make scalar field real
The simplest ansatz for complex scalar field:
φ(r) = φeiωt
This leads to Q-balls and boson stars solutions

Conclusion
The model for G = 0
SUSY potential U(|Φ|) = m2η2
susy 1 − exp −|Φ|2/η2
susy
Metric ds2 = −N(r)dt2 + 1
N(r) dr2 + r2 dθ2 + sin2
θdϕ2
with N(r) = 1 + r2
2 and = −3/Λ
Using Φ(t, r) = eiωt φ(r), rescaling
Equation of motion φ = −2
r φ − N
N φ − ω2
N2 φ + φ exp(−φ2)
N
Power law for symptotic fall-off for Λ < 0:
φ(r) = φ∆r∆, ∆ = −3
2 − 9
4 + 2
Charge and mass Q = 8π
∞
0 φr2dr and
M = 4π
∞
0 ω2φ2 + φ 2 + U(φ) r2dr

Conclusion
First results of the numerical analysis
ω
M
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
5010020050010002000
Mass over Omega
Λ
= 0
= −0.01
= −0.02
= −0.025
ω
M
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
5010020050010002000
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

Conclusion
φ(0)
M
0 2 4 6 8 10
110100100010000
Mass over Phi(0)
Λ
= 0
= −0.01
= −0.02
= −0.025
φ(0)
Q
0 2 4 6 8 10
110100100010000
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

Conclusion
M
Q
200 500 1000 2000 5000 10000 20000
200500200050002000050000
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
0246810
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) for
various values of Λ

Conclusion
M
Condensate
0 5000 10000 15000
0.0100.0150.0200.025
Condensate over Mass
Λ
= −0.03
= −0.04
= −0.05
= −0.075
Q
Condensate
0 5000 10000 15000 20000
0.0100.0150.0200.025
Condensate over Charge
Λ
= −0.03
= −0.04
= −0.05
= −0.075
Figure : Condensate O
1
∆ over Mass M (left) and charge Q (right) for various values
of Λ

Conclusion
φ(0)
Condensate
0 2 4 6 8 10
0.0100.0150.0200.025 Condensate over Phi(0)
Λ
= −0.03
= −0.04
= −0.05
= −0.075
Figure : Condensate O
1
∆ as function of the scalar ﬁeld at φ(0) for various values of Λ

Conclusion
SUSY potential U(|Φ|) = m2η2
susy 1 − exp −|Φ|2/η2
susy
The coupling constant κ is given with κ = 8πGη2
susy
Metric
ds2 = −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) = eiωt φ(r) and rescaling
Equations of motion
n = 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∆, ∆ = −3
2 − 9
4 + 2

Conclusion
Calculating the mass
φ(r) = φ∆r∆, ∆ = −3
2 − 9
4 + 2
The mass in the limit r 1 and κ > 0 is
n(r 1) = M + n1φ2
∆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

Conclusion
ω
M
0.2 0.4 0.6 0.8 1.0
10505005000
Mass over Omega
κ
= 0.0
= 0.001
= 0.01
= 0.05
= 0.1
ω
Q
0.2 0.4 0.6 0.8 1.0
10505005000
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) and
charge Q (right) versus frequency ω for various values of κ and ﬁxed Λ = 0.0

Conclusion
φ(0)
Q
0 2 4 6 8 10
10505005000
Charge over Phi(0)
κ
= 0.0
= 0.001
= 0.01
= 0.05
= 0.1
ω
φ(0)
0.2 0.4 0.6 0.8 1.0
051015
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

Conclusion
ω
Q
0.2 0.4 0.6 0.8 1.0
10505005000
Charge over Omega
κ
= 0.0
= 0.001
= 0.01
= 0.075
= 0.1
ω
Q
0.2 0.4 0.6 0.8 1.0
10505005000
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 versus
frequency ω for various values of κ and ﬁxed Λ = −0.001 (left) and ﬁxed Λ = −0.01
(right)

Conclusion
ω
φ(0)
0.2 0.4 0.6 0.8 1.0
05101520
Phi(0) over Omega
κ
= 0.0
= 0.001
= 0.01
= 0.075
= 0.1
ω
φ(0)
0.2 0.4 0.6 0.8 1.0
05101520
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 ω
for various values of κ and ﬁxed Λ = −0.001 (left) and ﬁxed Λ = −0.01 (right)

Conclusion
ω
Q
0.2 0.4 0.6 0.8 1.0 1.2 1.4
10505005000
Charge over Omega
Λ
= 0.0
= −0.001
= −0.01
= −0.05
= −0.1
ω
Q
0.2 0.4 0.6 0.8 1.0 1.2 1.4
10505005000
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 versus
frequency ω for various values of Λ and ﬁxed κ = 0.0 (left) and ﬁxed κ = 0.01 (right)

Conclusion
ω
φ(0)
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0246810
Phi(0) over Omega
Λ
= 0.0
= −0.001
= −0.01
= −0.05
= −0.1
ω
φ(0)
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0246810
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 ω
for various values of Λ and ﬁxed κ = 0.0 (left) and ﬁxed κ = 0.01 (right)

Conclusion
Summary of ﬁrst Results
Shift of ωmax for Q-balls and boson stars to higher values
for increasingly negative values of Λ, i.e.
ωmax → ∞ for Λ → −∞
The minimum value of the frequency for Q-balls is
ωmin = 0 for all Λ
The minimum value of the frequency for boson stars
ωmin increases for increasingly negative values of Λ
The curves mass M over frequency ω and charge Q
versus ω for Q-balls and boson stars show
M → 0 for ω → ωmax
Q → 0 for ω → ωmax

Conclusion
Summary of ﬁrst Results continued
For boson stars the cosmological constant Λ ’kills’ the
local 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. O
1
∆ as a
function of the scalar ﬁeld φ(0), has qualitatively the
same shape as in Horowitz and Way, JHEP 1011:011, 2010
[arXiv:1007.3714v2]

Conclusion
Outlook
Studying the condensate of boson stars in AdS with
SUSY potential
Interpreting the condensate in the context of CFT
Studying Q-balls and boson stars in AdS in (d+1)
dimensions
Studying rotating boson stars in AdS with SUSY
potential

SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time

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