SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time - Cancun talk 2012

Introduction
Q-balls
Boson stars
AdS/CFT correspondence
SUSY Q-balls in AdS background
SUSY boson stars in AdS background
Conclusion
SUSY Q-Balls and Boson Stars in Anti-de
Sitter space-time
Jürgen Riedel
in Collaboration with Betti Hartmann, Jacobs University Bremen
School of Engineering and Science
Jacobs University Bremen, Germany
CANCUN TALK 2012
Cancun, March 10th 2012
Jürgen Riedel & Betti Hartmann SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time

Introduction
Q-balls
Boson stars
Conclusion
Outline
1 Introduction
2 Q-balls
3 Boson stars
4 AdS/CFT correspondence
5 SUSY Q-balls in AdS background
6 SUSY boson stars in AdS background
7 Conclusion

Introduction
Q-balls
Boson stars
Conclusion
Solitons in non-linear field theories
General properties of soliton solutions
localized, finite energy, stable, regular solutions of
non-linear equations
can be viewed as models of elementary particles
dimension
Examples and restrictions
Skyrme model of hadrons in high energy physics one of
first models
Derrick’s theorem puts restrictions to localized soliton
solutions in more than one spatial dimension

Introduction
Q-balls
Boson stars
Conclusion
Solitons in non-linear field theories
Derrick’s non-existence theorem
Proof proceeds by contradiction
Suppose a solitonic solution φ0(x) exists
Deformations φλ(λx)=φ0(x), where λ is dilation parameter
No (stable) stationary point of energy exists with respect to
λ for a scalar with purely potential interactions.
Around Derrick’s Theorem
if one includes appropriate gauge fields, gravitational fields
or higher derivatives in field Lagrangian
if one considers solutions which are periodic in time

Introduction
Q-balls
Boson stars
Conclusion
Topolocial solitons
Properties
Boundary conditions at spatial inﬁnity are topological
different from that of the vacuum state
Degenerated vacua states at spatial inﬁnity
cannot be continuously deformed to a single vacuum
Example in one dimension: L = 1
2 (∂µφ)2
− λ
4 φ2 − m2
λ
broken symmetry φ → −φ with two degenerate vacua at
φ = ±m/
√
λ

Introduction
Q-balls
Boson stars
Conclusion
Non-topolocial solitons
Classical example in one dimension
With complex scalar ﬁeld
Φ(x, t) : L = ∂µΦ∂µΦ∗ − U(|Φ|), U(|Φ|) minimum at Φ = 0
Lagrangian is invariant under transformation
φ(x) → eiαφ(x)
Give rise to Noether charge Q = 1
i dx3φ∗ ˙φ − φ ˙φ∗)
Solution that minimizes the energy for ﬁxed Q:
Φ(x, t) = φ(x)eiωt

Introduction
Q-balls
Boson stars
Conclusion
Prominent examples for topological solitons
Further examples
vortices, magnetic monopoles, domain walls, cosmic
strings, textures
Prominent examples for non-topological solitons
Q-balls
Boson stars

Introduction
Q-balls
Boson stars
Conclusion
The model
Lagrangian L =∂µΦ∂µΦ∗ − U(|Φ|); the signature of the
metric is (+,-,-,-)
Noether current j = i(Φ∗ ˙Φ − Φ ˙Φ∗) symmetry under U(1)
Conserved Noether charge Q = 1
i d3(Φ∗ ˙Φ − Φ ˙Φ∗), with
Φ := Φ(t, r) we have dQ
dt = 0
Ansatz for solution Φ(x, t) = φ(x)eiωt
Energy-momentum tensor
Tµν = ∂µΦ∂νΦ∗ + ∂νΦ∂µΦ∗ − gµνL
Total Energy E = d3xT0
0 = d3x[| ˙Φ|2 + | Φ|2 + U(|Φ|)]
under assumption that gµν is time-independent

Introduction
Q-balls
Boson stars
Conclusion
Existence conditions of Q-balls
Condition 1
V (0) < 0; Φ ≡ 0 local maximum ⇒ ω2 < ω2
max ≡ U (0)
Condition 2
ω2 > ω2
min ≡ minφ[2U(φ)/φ2] minimum over all φ
Consequences
Restricted interval ω2
min < ω2 < ω2
max ;
U (0) > minφ[2U(φ)/φ2]
Q-balls are rotating in inner space with ω stabilized by
having a lower energy to charge ratio as the free particles

Introduction
Q-balls
Boson stars
Conclusion
Thin wall approximation of Q-balls
If the Q-ball is getting large enough, surface effects can
be ignored: thin wall limit.
Minimum of total energy ωmin = Emin = 2U(φ0)
φ2 , for φ0 > 0
The energy and charge is proportional to the volume
which is similarly found in ordinary matter → Q = ωφ2V
Therefore Q-balls in this limit are called Q-matter and have
very large charge, i.e. volume
Suitable potential U = m2|Φ|2 − a|Φ|4 + b|Φ|6, with a and
b are constants

Introduction
Q-balls
Boson stars
Conclusion
Rotating Q-balls
The Ansatz Φ = φ(r, θ)eiωt+inϕ, where n is an integer
Non-linear ﬁeld equation:
dU(φ)
dφ = ∂2φ
∂r2 + 2
r
∂φ
∂r + 1
r2
∂2φ
∂θ2 + cosθ
r2sinθ
∂φ
∂θ − n2φ
r2sinθ
+ ω2φ
Charge Q = 4πω
∞
0 drr2 π
0 dθsinθφ2
Uniqueness of the scalar ﬁeld under a complete
rotation Φ(ϕ) = Φ(ϕ + 2π) requires n to be an integer

Introduction
Q-balls
Boson stars
Conclusion
Rotating Q-balls
Consequences
The angular momentum J is quantized:
J = T0φd3x = nQ: n = rotational quantum number
One requires that φ →0 for r →0 or r → ∞
φ(r)|r=0 = 0 is a direct consequence of the term n2φ2
r2sin2θ

Introduction
Q-balls
Boson stars
Conclusion
Boson stars
Action ansatz: S =
√
−gd4x R
16πG + Lm
Matter Lagrangian Lm = −1
2 ∂µΦ∂µΦ∗ − U(|Φ|); the
signature of the metric is (-,+,+,+)
Variation with respect to the scalar ﬁeld
1√
−g
∂µ (
√
−g∂µΦ) = ∂U
∂|Φ|2 Φ
Metric ansatz
ds2 = −f(r)dt2 + l(r)
f(r) dr2 + r2dθ2 + r2sin2θdφ2
Conserved current jµ = i
√
−ggµν(Φ∗∂νΦ − Φ∂νΦ∗)
Noether charge Q = dx3j0 associated to the global
U(1) transformation

Introduction
Q-balls
Boson stars
Conclusion
Boson star models
Simplest model U = m2|Φ|2 (by Kemp, 1986)
Proper boson stars U = m2|Φ|2 − λ|Φ|4/2
(by Colpi, Sharpio and Wasserman, 1986)
Sine-Gordon boson star
U = αm2 sin(π/2 β |Φ|2 − 1 + 1
Cosh-Gordon boson star U = αm2 cosh(β |Φ|2 − 1
Liouville boson star U = αm2 exp(β2|Φ|2) − 1
(Schunk and Torres, 2000)

Introduction
Q-balls
Boson stars
Conclusion
Self-interacting boson stars models
Model U = m2|Φ|2 − a|Φ|4 + b|Φ|6, with a and b are
constants (Mielke and Scherzer, 1981)
Soliton stars U = m2|Φ|2 1 − |Φ|2/Φ2
0
2
(Friedberg, Lee and Pang, 1986)
Represented in the limit of ﬂat space − time, by Q -balls
as non-topological solitons
However, terms of |Φ|6 or higher-order terms implies that
the scalar part of the theory is not re-normalizable

Introduction
Q-balls
Boson stars
Conclusion
Charged Boson stars
System of complex scalar ﬁelds coupled to a
U(1) gauge ﬁeld with quartic self-interaction
The metric ansatz
ds2 = gµνdxµdxν = −A2Ndt2 + dr2
N + r2 dθ2 + sin2θdφ2 ,
with N = 1 − 2m(r)
r and
Solution ansatz: Φ = φ(r)eiωt , Aµdxµ = A0(r)dt
A gauge coupling constant e does increase the
maximum mass M and bf conserved charge Q
Using a V-shaped scalar potential
(Kleihaus, Kunz, Lammerzahl, and List, 2009)

Introduction
Q-balls
Boson stars
Conclusion
Rotating Boson stars
The metric ansatz
ds2 = −f(r, θ)dt2 +
l(r,θ)
f(r,θ)
g(r, θ)(dr2 + r2dθ2) + r2sin2θ dφ −
χ(r,θ)
r
dt
2
Stationary spherically symmetric ansatz
Φ(t, r, θ, ϕ) = φ(r, θ)eiωt+inϕ
Uniqueness of the scalar ﬁeld under a complete
rotation Φ(ϕ) = Φ(ϕ + 2π) requires n to be an integer (, i.e.
n = 0, ±1, ±2, ...)
Conserved scalar charge
Q = −4πω
∞
0
π
0
√
−g 1
f 1 + n
ω
χ
r φ2drdθ

Introduction
Q-balls
Boson stars
Conclusion
Rotating Boson stars continued
Total angular momentum J = − T0
ϕ
√
−gdrdϕdθ
With T0
ϕ = nj0, since ∂Φ
∂φ = i nΦ one ﬁnds: J = nQ
Solution is axially symmetric (for n = 0 )
This means that a rotating boson star is bf proportional to
the conserved Noether charge
If n = 0, it follows that a spherically symmetric boson
star has angular momentum J = 0
Rotating boson stars were intensively studied in 4
dimensions (Kleihaus et al) as well in 5 dimensions (Hartmann
et al) with U(|Φ|) = λ |Φ|6 − a|Φ|4 + b|Φ|2

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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 1.25010020050010002000
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

Introduction
Q-balls
Boson stars
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 10110100100010000
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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 200000.0100.0150.0200.025
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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 1.010505005000
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

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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 1.010505005000
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)

Introduction
Q-balls
Boson stars
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)

Introduction
Q-balls
Boson stars
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 1.410505005000
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)

Introduction
Q-balls
Boson stars
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)

Introduction
Q-balls
Boson stars
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

Introduction
Q-balls
Boson stars
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]

Introduction
Q-balls
Boson stars
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 - Cancun talk 2012

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