Integrability and Bethe Ansatz in the AdSCFT correspondence

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Integrability and Bethe Ansatz in the AdS/CFT
correspondence

• Konstantin Zarembo
• (Uppsala U.)

Thanks to Niklas Beisert (Princeton) Johan
Engquist (Utrecht) Gabriele Ferretti
(Chalmers) Rainer Heise (AEI, Potsdam) Vladimir
Kazakov (ENS) Andrey Marshakov (ITEP, Moscow) Joe
Minahan (Uppsala Harvard) Kazuhiro Sakai
(ENS) Sakura Schäfer-Nameki (Hamburg) Matthias
Staudacher (AEI, Potsdam) Arkady Tseytlin
(Imperial College Ohio State) Marija Zamaklar
(AEI, Potsdam)
Nordic Network Meeting Helsinki, 28.10.05
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AdS/CFT correspondence
Maldacena97
Gubser,Klebanov,Polyakov98 Witten98
3
Local operators and spin chains
related by SU(2) R-symmetry subgroup
j
i
i
j
4
Operator mixing
Renormalized operators
Mixing matrix (dilatation operator)
5
Multiplicatively renormalizable operators with
definite scaling dimension
anomalous dimension
6
Mixing matrix
Heisenberg Hamiltonian
7
Heisenberg model in Heisenberg representation
Heisenberg operators
Hiesenberg equations
8
Continuum classical limit
Landau-Lifshitz equation
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COMPARISON TO STRINGS
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( S5 fermions)
z
5D bulk
strings
0
gauge fields
4D boundary
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String theory in AdS5?S5
Metsaev,Tseytlin98

• Conformal 2d field theory (-function0)
• Sigma-model coupling constant
• Classically integrable

Classical limit is
Bena,Polchinski,Roiban03
12

• Need to know the spectrum of string states
• - eigenstates of Hamiltonian in light-cone
  gauge
• or
• - (1,1) vertex operators in conformal
  gauge
• Nothing of that is known
• But as long as ?gtgt1 semiclassical approximation
  is OK

Time-periodic classical solutions
Bohr-Sommerfeld
Quantum states
13
Consistent truncation
String on S3 x R1
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Conformal/temporal gauge
energy
2d principal chiral field well-known intergable
model
Pohlmeyer76 Zakharov,Mikhailov78 Faddeev,Resheti
khin86
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Equations of motion
Currents
Virasoro constraints
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Light-cone currents and spins
Classical spins
Virasoro constraints
Equations of motion
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High-energy approximation

Approximate solution at
The same (Landau-Lifshitz) equation describes the
spin chain in the classical limit!
Kruczenski03
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Integrability
Time-periodic solutions of classical equations of
motion
Spectral data (hyperelliptic curve meromorphic
differential)
AdS/CFT correspondence
Noether charges in sigma-model
Quantum numbers of SYM operators (L, M, ?)
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Global symmetries of the sigma-model
Left shifts
Right shifts
Time translations
World-sheet reparameterization invariance
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Noether charges
Length of the chain
Total spin
Energy (scaling dimension)
Virasoro constraints
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Dimensional analysis
Q any charge energy ? spins L, M
Dimensionless variables

• BMN coupling
• filling fraction

Berenstein,Maldacena,Nastase02
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BMN scaling
For any classical solution
Frolov-Tseytlin limit
If 1ltlt?ltltL2
Which can be compared to perturbation theory even
though ? is large.
Frolov,Tseytlin03
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String energy (strong-coupling calculation)
Anomalous dimension (weak-coupling calculation)

• three-loop discrepancy
• structural difference of finite-size/quantum
  corrections

Callan et al03 Beisert,Kristjansen,Staudacher03
Beisert,Dippel,Staudacher04
Beisert,Tseytlin05 Schäfer-Nameki,Zamaklar05
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Integrability
Equations of motion
Zero-curvature representation
equivalent
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Conserved charges
Generating function (quasimomentum)
on equations of motion
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Non-local charges
Local charges
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Auxiliary linear problem
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Dirac equation in 1d (j0, j1 are 2x2 matrices)
with spectral parameter x
Quasi-periodic boundary conditions
quasimomentum
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Noether charges
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Analytic structure of quasimomentum
p(x) is meromorphic on complex plane with cuts
along forbidden zones of auxiliary linear
problem and has poles at x1,-1
Resolvent
is analytic and therefore admits spectral
representation

and asymptotics at 8 completely
determine ?(x).
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Classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.04
Normalization
Momentum condition
Anomalous dimension
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Take
Normalization
Momentum condition
Anomalous dimension
This is the classical limit of Bethe equations
for spin chain!
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defined on cuts Ck in the complex plane
0
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(No Transcript)
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Exact quantum Bethe equations
In the scaling limit,
Taking the logarithm and expanding in 1/L
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Bethe equations for quantum strings?
Arutyunov,Frolov,Staudacher04 Staudacher04
Beisert,Staudacher05 Mann,Polchinski05 Ambjørn,J
anik,Kristjansen05
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Quantizing strings in AdS5xS5
Solving N4, D4 SYM at large N!
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IS N4 SYM SOLVABLE?
SPIN CHAINS
STRINGS
PLANAR DIAGRAMS
Universal relationship for large-N gauge theories?