Page 1
Integrability and Bethe Ansatz in the AdS/CFT correspondence
Konstantin Zarembo
(Uppsala U.)
Nordic Network MeetingHelsinki, 28.10.05
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)
Page 2
AdS/CFT correspondence Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Page 3
Local operators and spin chains
related by SU(2) R-symmetry subgroup
i j
i j
Page 4
Operator mixing
Renormalized operators:
Mixing matrix (dilatation operator):
Page 5
Multiplicatively renormalizable operators
with definite scaling dimension:
anomalous dimension
Page 6
Mixing matrix
Heisenberg Hamiltonian
Page 7
Heisenberg model in Heisenberg representation
Heisenberg operators:
Hiesenberg equations:
Page 8
Continuum + classical limit
Landau-Lifshitz equation
Page 9
COMPARISON TO STRINGS
Page 10
5D bulk
4D boundary
z
0
(+ S5 + fermions)
Page 11
String theory in AdS5S5Metsaev,Tseytlin’98
Bena,Polchinski,Roiban’03
• Conformal 2d field theory (¯-function=0)
• Sigma-model coupling constant:
• Classically integrable
Classical limit
is
Page 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 λ>>1 semiclassical approximation is OK
Time-periodic classical solutions
Quantum states
Bohr-Sommerfeld
Page 13
Consistent truncation
String on S3 x R1:
Page 14
Conformal/temporal gauge:
Pohlmeyer’76
Zakharov,Mikhailov’78
Faddeev,Reshetikhin’86
2d principal chiral field – well-known intergable model
~energy
Page 15
Equations of motion
Currents:
Virasoro constraints:
Page 16
Light-cone currents and spins
Virasoro constraints:
Classical spins:
Equations of motion:
Page 17
High-energy approximation
Approximate solution at :
The same (Landau-Lifshitz) equation describes
the spin chain in the classical limit!Kruczenski’03
Page 18
Integrability:
AdS/CFT correspondence:
Time-periodic solutions of classical equations of motion
Spectral data (hyperelliptic curve + meromorphic differential)
Noether charges in sigma-model
Quantum numbers of SYM operators (L, M, Δ)
Page 19
Global symmetries of the sigma-model
Left shifts:
Right shifts:
Time translations:
World-sheet reparameterization invariance
Page 20
Noether charges
Length of the chain:
Total spin:
Energy (scaling dimension):
Virasoro constraints:
Page 21
“Dimensional analysis”
Q – any charge: energy Δ; spins L, M; …
Dimensionless variables:
• BMN coupling:
• filling fraction:
Berenstein,Maldacena,Nastase’02
Page 22
BMN scaling
Frolov,Tseytlin’03
For any classical solution:
Frolov-Tseytlin limit:
If 1<<λ<<L2:
Which can be compared to perturbation theory even
though λ is large.
Page 23
• three-loop discrepancy
• structural difference of finite-size/quantum corrections
String energy (strong-coupling calculation):
Anomalous dimension (weak-coupling calculation):
Callan et al’03; Beisert,Kristjansen,Staudacher’03; Beisert,Dippel,Staudacher’04
Beisert,Tseytlin’05; Schäfer-Nameki,Zamaklar’05
Page 24
Integrability
Zero-curvature representation:
Equations of motion:
equivalent
Page 25
Conserved charges
time
on equations of motion
Generating function (quasimomentum):
Page 26
Non-local charges:
Local charges:
Page 27
Auxiliary linear problem
Page 28
quasimomentum
Dirac equation in 1d (j0, j1 are 2x2 matrices) with
spectral parameter x
Quasi-periodic boundary conditions:
Page 30
Analytic structure of quasimomentum
p(x) is meromorphic on complex plane with cuts along
forbidden zones of auxiliary linear problem and has poles
at x=+1,-1
Resolvent:
is analytic and therefore admits spectral representation:
and asymptotics at ∞
completely determine ρ(x).
Page 31
Classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.’04
Normalization:
Momentum condition:
Anomalous dimension:
Page 32
Normalization:
Momentum condition:
Anomalous dimension:
Take
This is the classical limit of Bethe equations for spin chain!
Page 33
defined on cuts Ck in the complex plane
x
0
Page 35
In the scaling limit,
Taking the logarithm and expanding in 1/L:
Exact quantum Bethe equations:
Page 36
Bethe equations for quantum strings?
Arutyunov,Frolov,Staudacher’04
Staudacher’04; Beisert,Staudacher’05
Mann,Polchinski’05
Ambjørn,Janik,Kristjansen’05
Page 37
Quantizing strings in AdS5xS5
Solving N=4, D=4 SYM at large N!
Page 38
PLANAR DIAGRAMS SPIN CHAINS STRINGS
IS N=4 SYM SOLVABLE?
Universal relationship for large-N gauge theories?