Bethe Ansatz in AdS/CFT Correspondence

1
Bethe Ansatz in AdS/CFT Correspondence

• Konstantin Zarembo
• (Uppsala U.)

J. Minahan, K. Z., hep-th/0212208 N. Beisert, J.
Minahan, M. Staudacher, K. Z., hep-th/0306139 V.
Kazakov, A. Marshakov, J. Minahan, K. Z.,
hep-th/0402207 N. Beisert, V. Kazakov, K. Sakai,
K. Z., hep-th/0503200 N. Beisert, A. Tseytlin, K.
Z., hep-th/0502173 S. Schäfer-Nameki, M.
Zamaklar, K.Z., hep-th/0507179
DGMTP, Tianjin, 23.08.05
2
Large-N expansion of gauge theory
String theory
Early examples

• 2d QCD
• Matrix models

4d gauge/string duality

• AdS/CFT correspondence

3
Macroscopic strings from planar diagrams
Large orders of perturbation theory
Large number of constituents
or
4
AdS/CFT correspondence
Maldacena97
Gubser, Klebanov, Polyakov98 Witten98
5
Quantum string
?ltlt1
Strong coupling in SYM
Classical string
Way out consider states with large quantum
numbers operators with large number of
constituent fields
Price highly degenerate operator mixing
6
Operator mixing
Renormalized operators
Mixing matrix (dilatation operator)
Multiplicatively renormalizable operators with
definite scaling dimension
anomalous dimension
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N4 Supersymmetric Yang-Mills Theory

• Field content

The action
8
Local operators and spin chains

• Restrict to SU(2) sector

related by SU(2) R-symmetry subgroup
a
b
b
a
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Operator basis

• 2L degenerate operators
• The space of operators can be identified with the
  Hilbert space of a spin chain of length L
  with (L-M) ?s and M
  ?s

10
One loop planar (N?8) diagrams
11
Permutation operator

• Integrable Hamiltonian! Remains such
• at higher orders in ?
• for all operators

Beisert, Kristjansen, Staudacher03 Beisert,
Dippel, Staudacher04
Beisert, Staudacher03
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Spectrum of Heisenberg ferromagnet
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Ground state
(SUSY protected)
Excited states
flips one spin
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Non-interacting magnons

• good approximation if MltltL

• Exact solution
• exact eigenstates are still multi-magnon Fock
  states
• () stays the same
• but () changes!

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Bethe ansatz
Rapidity
Bethe31
Zero momentum (trace cyclicity) condition
Anomalous dimension
16
bound states of magnons Bethe strings
0
mode numbers
17
Macsoscopic spin waves long strings
Sutherland95 Beisert, Minahan, Staudacher,
K.Z.03
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Scaling limit
defined on cuts Ck in the complex plane
0
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Classical Bethe equations
Normalization
Momentum condition
Anomalous dimension
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Comparison to strings

• Need to know the spectrum of string states
• - eigenstates of Hamiltonian in light-cone
  gauge
• or
• - (1,1) vertex operators in conformal
  gauge
• Not known how to quantize strings in AdS5xS5
• But as long as ?gtgt1 semiclassical approximation
  is OK

Time-periodic classical solutions
Bohr-Sommerfeld
Quantum states
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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
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Consistent truncation
Keep only
String on S3xR1
Conformal/temporal gauge
2d principal chiral field well-known intergable
model
Pohlmeyer76 Zakharov, Mikhailov78 Faddeev,
Reshetikhin86
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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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Noether charges
Length of the chain
Total spin
Energy (scaling dimension)
Virasoro constraints
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BMN scaling
BMN coupling
Berenstein, Maldacena, Nastase02
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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Integrability
Equations of motion
Zero-curvature representation
equivalent
on equations of motion
Infinte number of conservation laws
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Auxiliary linear problem
quasimomentum
Noether charges are determined by asymptotic
behaviour of quasimomentum
28
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, K.Z.04
Normalization
Momentum condition
Anomalous dimension
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Take
Normalization
Momentum condition
Anomalous dimension
This is classical limit of Bethe equations for
spin chain!
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Q Can we quantize string Bethe equations
(undo thermodynamic limit)? A Yes! Arutyunov,
Frolov, Staudacher04 Staudacher04Beisert,
Staudacher05

• Quantum strings in AdS
• BMN limit
• Near-BMN limit
• Quantum corrections to classical string solutions

Berenstein, Maldacena, Nastase02 Metsaev02
Callan, Lee,McLoughlin,Schwarz,Swanson,Wu03
Frolov, Tseytlin03 Frolov, Park,
Tsetlin04 Park, Tirziu, Tseytlin05 Fuji,
Satoh05
Finite-size corrections to Bethe ansatz
Beisert, Tseytlin, Z.05 Hernandez, Lopez,
Perianez, Sierra05 Schäfer-Nameki, Zamaklar,
Z.05
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String on AdS3xS1
angle in AdS
angle on S5
radial coordinate in AdS
Rigid string solution
Arutyunov, Russo, Tseytlin03
AdS spin
angular momentum on S5
One-loop quantum correction
Park, Tirziu, Tseytlin05
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Bethe equations
Even under L?-L
First correction is O(1/L2)
But singular if
simultaneously
Local anomaly
Kazakov03

• cancels at leading order
• gives 1/L correction

Beisert, Kazakov, Sakai, Z.05
Beisert, Tseytlin, Z.05 Hernandez, Lopez,
Perianez, Sierra05
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0
Locally
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Anomaly
local contribution
1/L correction to classical Bethe equations
Beisert, Tseytlin, Z.05
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Re-expanding the integral
Agrees with the string calculation.

• Remarks
• anomaly is universal depends only on singular
  part
• of Bethe equations, which is always the same
• finite-size correction to the energy can be
  always
• expressed as sum over modes of small
  fluctuations

Beisert, Freyhult05