Spin chain from marginally
deformed AdS3 x S3
Wen-Yu Wen @ NTU
hep-th/0610147
hep-th/0702xxx
string/spin chain correspondence
classical spinning string on S3
FAST STRING LIMIT
M. Kruczenski, Phys. Rev. Lett. 93
SU(2) Heisenberg spin chain XXX1/2
Landau-Lifshitz equation
AdS3/SL(2) S.Bellucci, P.Y.Casteill, J.F.Morales and C.Sochichiu, Nucl. Phys. B 707
SU(3) R.~Hernandez and E.~Lopez, JHEP 0404
string/spin chain correspondence
classical spinning string on deformed S3
FAST STRING LIMIT
+
SMALL DEFORMATION LIMIT
anisotropic SU(2) Heisenberg spin chain XXY1/2
generalized Landau-Lifshitz equation
TARGET SPACE
• start with AdS5 x S5 background
k
ds = ( − cosh 2 ρdt 2 + dρ 2 + sinh 2 ρdΩ3 )
4
k
~
2
2
2
2
+ (dθ + sin θdφ + cos θdΩ3 )
4
2
• marginally deformed S3 from WZW sigma model + marginal operator
D.Israel, C.Kounnas, D.Orlando and P.M.Petropoulos, Fortsch. Phys. 53
• 1-parameter deformed U(1) fiber in Hopf fibration of S3.
S3
• to be explicit,
H
U(1)
S2
~
dΩ 3 = dβ 2 + sin 2 β dα 2 + (1 − 2 H 2 )(dγ + cos β dα ) 2
STRING WORLDSHEET
• embedding
ρ = θ = 0 α (τ , σ ) β (τ , σ ) γ (τ , σ )
• gauge fixing
• spinning
t = κτ
α →α +t
•induced Polyakov action
λk
2
2
2
2
2
2
2
2
2
′
′
′
d
τ
d
σ
H
βκ
H
β
α
β
H
γ
2
cos
(
1
2
cos
)
(
1
2
)
−
−
−
−
−
−
16π ∫∫
− 2(1 − 2 H 2 ) cos β α ′γ ′
S=
+ 2(1 − 2 H 2 cos 2 β )κα&+ 2(1 − 2 H 2 ) cos βκγ&
+ (1 − 2 H 2 cos 2 β )α&2 + β&2 + (1 − 2 H 2 )γ&2 + 2(1 − 2 H 2 ) cos β α&γ&
• FAST STRING LIMIT
κ → ∞ X&μ → 0 κX μ ′ finite
STRING WORLDSHEET
• virasora constraints
2(1 − 2 H 2 cos 2 β )κα ′ + 2(1 − 2 H 2 ) cos βκγ ′ = 0
• resulting action
λk
2
2
2
2
2
′
′
S=
d
τ
d
σ
−
H
βκ
−
β
−
Δ
β
γ
2
cos
(
)
H
16π ∫∫
+ 2(1 − 2 H 2 cos 2 β )κα&+ 2(1 − 2 H 2 ) cos βκ γ&
(1 − 2 H 2 ) sin 2 β
Δ H (β ) ≡
1 − 2 H 2 cos 2 β
Hamiltonian density
conjugated angular momentum density
H=0
• no deformation
Δ H = sin 2 β
• Hamiltonian density
ρ ρ
H = β ′ + sin β γ ′ = S ′ ⋅ S ′
2
2
2
ρ
S = (sin β cos γ , sin β sin γ , cos β )
• conjugate angular momentum
λk
πα =
κ ∫ dσ = J 0
8π
λk
πγ =
κ ∫ dσ cos β
8π
length of spin chain
Wess-Zumino term
H≠0
• Hamiltonian density
H = β ′ 2 + Δ H ( β )γ ′ 2 + 2κ 2 H 2 cos 2 β
ρ ρ
~ ~
~
≈ S ′ ⋅ S ′ + ( H ⋅ S z )2
~ρ
~
S = ( Δ H cos γ , Δ H sin γ , S z )
satisfying uniform speed condition
• conjugate angular momentum
J = J 0 + 2κH 2 ∫ dσ cos 2 β
′
~′
( Δ H )2 + ( S z )2 = 1
• no analytic solution for S were found in deformed background
hyperbolic deformed AdS3
deformed S3
Z
4
Z
3.5
1
3
2.5
0.5
2
-1
-0.5
0.5
1
1.5
Y
-2
-0.5
-1
-1
1
0.5
2
Y
SMALL DEFORMATION LIMIT
• to render analytic solution for S, but still keep nontrivial
interaction term
H → 0 κH finite
• the resulting action
λk
2
2
2
2
2
2
′
′
2
cos
sin
τ
σ
−
βκ
−
β
−
β
γ
S=
d
d
H
16π ∫∫
+ 2κα&+ 2 cos βκγ&
• Hamiltonian density indicates the anisotropic spin chain system
ρ ρ
2
2
′
′
H = S ⋅ S + 2H Sz
Landau-Lifshitz equation
• equation of motion
β ′′ − sin β γ&− sin β cos β (γ ′ ) 2 + H 2 sin 2β = 0
sin β β&+ (sin 2 β γ ′ )′ = 0
can also be written as (generalized) Landau-Lifshitz equation
ρ ρ 2ρ ρ ρ
∂ t S = S × ∂ σ S + S × ☺S
⎛ j1
⎜
☺= ⎜
⎜
⎝
with
j2
⎞
⎟
⎟
j3 ⎟⎠
j1 = j2 = 1
j3 = 1 − 2 H 2
Conclusion
• Landau-Lifshitz equation is an integral system, which implies
at least a sector of string theory in AdS3xS3 is also integrable.
• In principle, we may work out small deformation limit of deformed
AdS3 and explicitly show there is also an noncompact version of
LL equation.
• We may also work backwards, to find out which spinning string
solution corresponds to most general LL equation with all j’s
nontrivial.
• If we like, we may also include the interaction term which is linear
to Sz.
• It is interesting to see if similar correspondence exists in higher
dimensions.
Thank You