vortex solutions on membranes · introduction ads/cft ads/cft correspondence (some) strongly...
TRANSCRIPT
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Vortex solutions on membranes
Vortex solutions on membranes
Shinsuke Kawai
SKKU, Suwon
APCTP YongPyong Winter School, February 2010
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Vortex solutions on membranes
Outline
1 IntroductionAdS/CFTAdS/CMTThe ABJM model
2 Non-relativistic mass-deformed ABJMRelativistic mass-deformed ABJMThe non-relativistic limitNon-relativistic SUSY
3 BPS vortex solutionsThe BPS equationsThe exact vortex solutionsExamplesPreserved supersymmetry
4 Summary and comments
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Vortex solutions on membranes
Introduction
Introduction
What is string theory?
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Vortex solutions on membranes
Introduction
Introduction
What is string theory?
The theory of strong interaction
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Vortex solutions on membranes
Introduction
Introduction
What is string theory?
The theory of strong interaction
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Vortex solutions on membranes
Introduction
Introduction
What is string theory?
The theory of strong interaction
The theory of everything
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Vortex solutions on membranes
Introduction
Introduction
What is string theory?
The theory of strong interaction
The theory of everything
AdS/CFT: Logical completion of QFT(incl. QCD, Condensed Matter Systems...)
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Vortex solutions on membranes
Introduction
AdS/CFT
AdS/CFT correspondence
(Some) strongly coupled field theories have gravity duals [Maldacena]
Conformal group SO(d , 2) ↔ AdSd+1
(Poincare Mµν ,Pµ, Dilatation D, SCG Kµ)
Poincare coordinates are
ds2 = ρ2ηµνdxµdxν +dρ2
ρ2
Dilatation is D : xµ 7→ λxµ, ρ 7→ ρ/λ
Dictionary: [Gubser, Klebanov, Polyakov] [Witten]
〈eR
φ0O〉 = ZAdS [φ0] ∼ e−S[φ0]
Boundary conditions ↔ statesVariation of boundary conditions ↔ correlation functions
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Vortex solutions on membranes
Introduction
AdS/CFT
Radial coordinate ↔ energy scale (‘holographic renormalization’)
A realisation of holographic principle
Canonical example:AdS5 × S5 in Type II B string theory ↔ N = 4 SYM in 4 dimSO(4, 2)conf × SO(6)R : isometries of the spacetime
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Vortex solutions on membranes
Introduction
AdS/CFT
λ = g 2YMN : ’t Hooft coupling
3 versions of AdS/CFT
weak: large λ
strong: any finite λ but N → ∞ and gs = g 2YM → 0
(exact in α′ but for small gs only)
strongest: any gs and N
(α′ and gs)
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Vortex solutions on membranes
Introduction
AdS/CMT
Condensed matter applications of AdS/CFT
Application to QCD relatively successful– why not Condensed Matter Theory?
AdS/CFT in laboratory
Strongly correlated systems near criticality in low dimensionsRich examplesGood control
A new computational tool in CMT
Beyond perturbative field theory and lattice
A new arena of string theory
Typically, non-relativistic (∆ ∼ (q − qc)νz , ξ ∼ (q − qc)
−ν)D : t 7→ λz t, x i 7→ λx i
NR AdS/CFT duality: limited technology – a new challenge
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Vortex solutions on membranes
Introduction
AdS/CMT
Examples of condensed matter systems
SuperconductivitySuperfluidityQuantum Hall systems
AdS/CFT involves large N SYM.
The AdS is realised by some other fields – CMT is assumed tobe a probe on a background geometrySupersymmetry – broken by finite T (?)Phenomenological approach – be maximally optimistic and useas a computational tool
Questions
Phase transition, solitonic excitation (brane excitations)Especially, vortices play important roles in above examplesABJM: M-theory on AdS4 × S7/Zk ⇐⇒ CFT on M2-branes
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Vortex solutions on membranes
Introduction
AdS/CMT
Superfluidity in He
4He phase diagram (left) and 3He phase diagram (right)
Source: Low Temperature Laboratory, TKK, Finland
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Vortex solutions on membranes
Introduction
AdS/CMT
Vortices in 3He superfluidity
A closer look at the low temperature 3He phase diagram (left) andtwo types of vortices in the superfluid B phase (right)
Source: Low Temperature Laboratory, TKK, Finland
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Vortex solutions on membranes
Introduction
AdS/CMT
Our focus: vortex solutions on the CFT side.Solitonic solutions in ABJM
Abelian vortices (Jackiw-Lee-Weinberg type) and domain walls[Arai, Montonen, Sasaki 2008]
Non-abelian vortices [Kim, Kim, Kwon, Nakajima 2009] [Auzzi, Kumar 2009]
Abelian, non-relativistic vortices [Kawai, Sasaki 2009]
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Vortex solutions on membranes
Introduction
The ABJM model
The ABJM model
The ABJM model [Aharony, Bergman, Jafferis, Maldacena (2008)]
Sbos
ABJM =
∫
d3x(
Lbos
kin + LCS − V bos
D − V bos
F
)
,
Lbos
kin= −Tr
h
(DµZA)†(D
µZ
A) + (DµWA)
†(D
µWA)
i
,
LCS =k
4πǫµνλ
Tr
h
Aµ∂νAλ +2i
3AµAνAλ − Aµ∂ν Aλ −
2i
3AµAν Aλ
i
,
Vbos
D =4π2
k2Tr
h
˛
˛
˛
˛
ZB
Z†B
ZA− Z
AZ†B
ZB
− W†B
WBZA
+ ZAWBW
†B˛
˛
˛
˛
2
+
˛
˛
˛
˛
W†B
WBW†A
− W†A
WBW†B
− ZB
Z†B
W†A
+ W†A
Z†B
ZB
˛
˛
˛
˛
2 i
,
Vbos
F =16π2
k2Tr
h
˛
˛
˛
˛
ǫACǫBD
WBZC
WD
˛
˛
˛
˛
2
+
˛
˛
˛
˛
ǫACǫBD
ZB
WC ZD
˛
˛
˛
˛
2 i
.
Aµ, Aµ · · ·U(N)×U(N) gauge fields, and Z A, W †A (A = 1, 2, A = 3, 4)
· · · complex scalars in U(N) × U(N) bi-fundamental (N, N) rep
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Vortex solutions on membranes
Introduction
The ABJM model
Chern-Simons-matter theory on 2+1 dimensions
2+1 dim
X1,...,X8
(Z A,W†A) → e2πik (Z A,W†A)
M-theory on AdS4 × S7/Zk
N, k → ∞, λ = Nk
fixed (’t Hooft limit): IIA on AdS4 × CP3
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Vortex solutions on membranes
Introduction
The ABJM model
We consider:The ABJM model (Chern-Simons-matter theory)
↓ massive deformationMass-deformed ABJM
↓ non-relativistic limitNR, mass-deformed ABJM
↓ solving BPS eqnsBPS vortex solutions
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Vortex solutions on membranes
Introduction
The ABJM model
A bigger picture:
ABJM CSm-th
��
ks +3 M-theory on AdS4 × S7/Zk
��
Massive ABJM
��
Fuzzy spheres
��
NR Massive ABJM
��
NR
��
NR BPS vortices ks +3 gravity dual of NR BPS vortices?
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Vortex solutions on membranes
Non-relativistic mass-deformed ABJM
Relativistic mass-deformed ABJM
Relativistic mass-deformed ABJM
Before taking the NR limit we take massive deformation[Hosomichi, Lee3, Park 2008], [Gomis, Rodrıguez-Gomez, Van Raamsdonk, Verlinde 2008]
· · · introducing a scale that is necessary for the solitonic solutionsMaximally supersymmetric (N = 6) massive deformation
SO(8)R → SU(2) × SU(2) × U(1) × Z2
The scalars acquire equal masses
The change of (the bos. part of) the Lanrangian:
δL = Tr
[
− m2Z†AZ A − m2W †AWA
+ 4πmk
(
(Z AZ†A)2 − (W †AWA)2 − (Z †
AZ A)2 + (WAW †A)2
)
]
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Vortex solutions on membranes
Non-relativistic mass-deformed ABJM
The non-relativistic limit
The non-relativistic limit
Now we consider the NR limit [Nakayama, Sakaguchi, Yoshida 2009], [Lee3 2009]
1 Write down the action with c and ~ explicitly
2 Decompose: Z A = ~√2m
(
e−i mc2 t~ z A + e i mc2t
~ z∗A)
etc.
(z A, z∗A are non-relativistic scalar fields)
Keep the particle DoF (z A,w †A) & drop the antiparticles
3 Send c , m → ∞ and look at the leading orders
The resulting (bos. part of the) Lagrangian:
LNR,bos
ABJM=
k~c
4πǫµνλ
Tr
»
Aµ∂νAλ +2i
3AµAνAλ − AµAν Aλ −
2i
3AµAν Aλ
–
+Tr
"
i~
2
„
−z†A
Dt zA
+ Dt zA· z
†A
«
−~
2
2mDi z
ADi z
†A
+i~
2
“
−wADtw†A
+ Dtw†A
· wA
”
−~
2
2mDi w
†ADi wA
+π~
2
km
(zAz†A
)2− (z
†A
zA)2− (w
†AwA)
2+ (wAw
†A)2
ff
#
.
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Vortex solutions on membranes
Non-relativistic mass-deformed ABJM
The non-relativistic limit
The equations of motion
The scalar part: nonlinear Schrodinger equations
i~Dt zA
= −~
2
2mD
2i z
A−
2π~2
km(z
Bz†B
zA− z
Az†B
zB
),
i~Dtw†A
= −~
2
2mD
2i w
†A+
2π~2
km(w
†BwBw
†A− w
†AwBw
†B).
The gauge field part: Gauss-law constraints
The fermionic part:
i~Dtψ−A + 2mc2δ
AAψ−A
− i~cD−ψ+A = 0,
i~Dtψ+A + 2mc2δ
AAψ+A
− i~cD+ψ−A = 0.
(due to these NR Dirac eqns 1/2 of the fermionic DoF drop)
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Vortex solutions on membranes
Non-relativistic mass-deformed ABJM
Non-relativistic SUSY
Non-relativistic SUSY
Apply the same procedure of NR limit to SUSY trfn rules→ non-relativistic SUSY (super Schrodinger symmetry)14 supercharges:
10 kinematical SUSY:(
ω+AB, ω−AB , ω±AB
)
2 dynamical SUSY:(
ω−AB, ω+AB
)
2 conformal SUSY:(
ξAB, ξAB
)
SUSY parameters defined by
ωAB = ǫi ΓiAB (i = 1, 2, ..., 6), ω = 1√
2
„
ω− + ω+−iω− + iω+
«
, ωAB± = (ω±AB )† = 1
2ǫABCD ω±CD
Conformal SUSY ∼ Special conformal charge × Dynamical SUSY,S = i [K ,QD ]
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Vortex solutions on membranes
BPS vortex solutions
The BPS equations
The BPS equations
The Hamiltonian density (Noether charge of the time translation)
H = Tr
"
~2
2m|Di z
A|2
+~
2
2m|Di w
†A|2−π~
2
km
(zAz†A
)2− (z
†A
zA)2− (w
†AwA)
2+ (wAw
†A)2
ff
#
.
Using D± ≡ D1 ± iD2 and Bogomol’nyi completion =⇒ BPS bound:
E =
∫
d2x H =
∫
d2x Tr
[
~2
2m
∣
∣
∣D−z A
∣
∣
∣
2
+~2
2m
∣
∣
∣D+w †A
∣
∣
∣
2]
≥ 0
saturated by BPS equations
D−z A = 0, D+w †A = 0,
Recall: z A and w †A are matrix-valued (N × N)
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Vortex solutions on membranes
BPS vortex solutions
The BPS equations
The fuzzy 3-sphere ansatz [Gomis, Rodrıguez-Gomez, Van Raamsdonk, Verlinde 2008]
z A(x) = ψz(x)S I ,w †A(x) = ψw (x)S I ,Ai (x) = ai (x)S IS†I , Ai (x) = ai (x)S†
I S I .
Here, ψz , ψw , ai ∈ C, (S†1 )mn =
√m − 1δmn, (S†
2 )mn =√
N − mδm+1,n
SI
= SJS†J
SI− S
IS†J
SJ,
S†I
= S†I
SJS†J
− S†J
SJS†I,
Tr SIS†I
= Tr S†I
SI
= N(N − 1).
Then the BPS eqns =⇒ the Jackiw-Pi vortex eqns [Jackiw and Pi 1990]
(D1 − iD2)ψz(x) = 0, (D1 + iD2)ψw (x) = 0 (Di ≡ ∂i + iai)
J-P eqns allow exact solutions
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Vortex solutions on membranes
BPS vortex solutions
The exact vortex solutions
The exact vortex solutions
Finding solutions:Set w †A = 0 & solve for z A, Ai and Ai (BPS I), orset z A = 0 & solve for w †A, Ai and Ai (BPS II)
The radius of the fuzzy S3 (in the case of BPS I):
R2 =2
NTM2Tr
[
Z AZ†A
]
=N − 1
TM2
|ψz |2m
,
TM2: the tension of an M2-brane
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Vortex solutions on membranes
BPS vortex solutions
The exact vortex solutions
The BPS I solutions
1 Changing the variables ψz(x) = e iθ(x)ρ12 (x), (θ, ρ ∈ R),
the BPS eqns become
ai (x) = −∂iθ +1
2ǫij∂
j ln ρ.
2 Using the Gauss law constraint=⇒ Liouville equation ∇2 lnρ = −4π
kρ,
solved by
ρ(x) =k
2π∇2 ln
(
1 + |f (z)|2)
(f (z): a holomorphic function of z = x1 + ix2)
3 θ is fixed by regularity of ψz at z = 0
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Vortex solutions on membranes
BPS vortex solutions
Examples
Examples
Examples of BPS I solutionsChoose a profile: f (z) =
(
z0
z
)n, n ∈ Z, z0: a complex const; yielding
ρ(x) =k
2π
4n2
r20
“
rr0
”2(n−1)
»
1 +“
rr0
”2n–2, θ = −(n − 1) arg z = −(n − 1) arctan(x2/x1)
These are non-topological vortices since |ψz | → 0 as |z | → ∞
-2
-1
0
1
2
x1-2
-1
0
1
2
x2
00.20.40.60.8Ρ
-2
-1
0
1
2
x1
-2
-1
0
1
2
x1-2
-1
0
1
2
x2
00.20.40.60.8Ρ
-2
-1
0
1
2
x1
-2
0
2x1
-2
-1
0
1
2
x2
0
0.5
1Ρ
-2
0
2x1
Figure: |ψz |2 shown for f (z) = 1z, 1
z2 and 1z(z−1) , with k = 1
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Vortex solutions on membranes
BPS vortex solutions
Examples
Examples of BPS II solutions
Setting z A = 0 we find similar solutions for w †A:
ψw (x) = e iθ(x)ρ12 (x),
ρ(x) = − k
2π∇2 ln
(
1 + |f (z)|2)
,
θ = (n − 1) arctan(x2/x1).
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Vortex solutions on membranes
BPS vortex solutions
Preserved supersymmetry
Preserved supersymmetry
Check the SUSY trfns against the BPS eqns (in the BPS-I case)
w †A = 0, D−z A = 0.
The fermion transformation rules:
δKψ+A= −ω
+ABzB, δKψ−A
= +ω−ABzB,
δDψ+A=
i
2mω−AB
D+zB, δDψ−A = 0,
δSψ+A∼ ξ
ABzB, δSψ−A
∼ 0.
Hence δψ = 0 =⇒ ω+AB = ω−AB = ω−AB = ξAB
= 0
This means that the BPS-I solutions break 5 kinematical, 1dynamical and 1 conformal SUSYs (i.e. exactly 1/2).
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Vortex solutions on membranes
BPS vortex solutions
Preserved supersymmetry
The BPS II case is similar.Summarising the results,
Type of Kinematical Dynamical ConformalSUSY ω+AB ω+AB ω−AB ω−AB ω−AB ω+AB ξ
ABξAB
BPS I © × © × × © × ©BPS II × © × © © × © ×
Table: ©: preserved, ×: broken
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Vortex solutions on membranes
BPS vortex solutions
Preserved supersymmetry
Summary of our solutions
We find exact solutions of abelian vortices by solving BPSequations in the non-relativistic ABJM model: Jackiw-Picombined with fuzzy 3-sphere
These solutions preserve half of the super Schrodingersymmetry
Any relevance in real physics?– more realistic, parity broken models with external fieldsdesirable.
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Vortex solutions on membranes
BPS vortex solutions
Preserved supersymmetry
Vortex solutions in AdS
Vortex line
Pure AdS [Dehghani Ghezelbash Mann, 2001]
AdS-Sch [Dehghani Ghezelbash Mann, 2001]
with boundary magnetic field
[Albash Johnson 2009]
[Montull Pomarol Silva 2009]
[Maeda Natuume Okamura 2009]
with vanishing magnetic field on the boundary[Keranen Keski-Vakkuri Nowling Yogendran 2009]
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Vortex solutions on membranes
Summary and comments
Unsorted list of problems
Non-relativistic AdS/CFT exists at all?
In 2 + 1 dim, spontaneous breaking of continuous symmetry isnot possible at finite temperature (Mermin-Wagner).However, such a phase transition is found in holographicsuperconductor. How do we interpret? Large N artefact?
Berezinskii-Kosterlitz-Thouless transition in AdS?