Bethe ansatz in String Theory

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Bethe ansatz in String Theory

• Konstantin Zarembo
• (Uppsala U.)

Integrable Models and Applications, Lyon,
13.09.2006
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AdS/CFT correspondence
Maldacena97
Gubser,Klebanov,Polyakov98 Witten98
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Planar diagrams and strings
time
(kept finite)
t Hooft coupling String coupling constant
(goes to zero)
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Strong-weak coupling interpolation
?
0
SYM perturbation theory
String perturbation theory
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Circular Wilson loop (exact)
Erickson,Semenoff,Zarembo00 Drukker,Gross00
Minimal area law in AdS5
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Weakly coupled SYM is reliable if
Weakly coupled string is reliable if
Can expect an overlap.
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N4 Supersymmetric Yang-Mills Theory
Gliozzi,Scherk,Olive77

• Field content

Action
Global symmetry PSU(2,24)
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Spectrum
Basis of primary operators
Spectrum ?n
Dilatation operator (mixing matrix)
10
Local operators and spin chains
related by SU(2) R-symmetry subgroup
b
a
a
b
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Tree level ?L (huge degeneracy)
One loop
Minahan,Z.02
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Bethe ansatz
Bethe31
Zero momentum (trace cyclicity) condition
Anomalous dimensions
13
Higher loops
Requirments of integrability and BMN
scaling uniquely define perturbative scheme to
construct dilatation operator through order
?L-1
Beisert,Kristjansen,Staudacher03
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The perturbative Hamiltonian turns out to
coincide with strong-coupling expansion of
Hubbard model at half-filling
Rej,Serban,Staudacher05
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Asymptotic Bethe ansatz
Beisert,Dippel,Staudacher04
In Hubbard model, these equations are
approximate with O(e-f(?)L) corrections at L?8
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Anti-ferromagnetic state
Rej,Serban,Staudacher05 Z.05 Feverati,Fiorovan
ti,Grinza,Rossi06 Beccaria,DelDebbio06
Weak coupling
Strong coupling
Q Is it exact at all ??
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Arbitrary operators
Bookkeeping
letters
words
sentences
Spin chain
infinite-dimensional representation of PSU(2,24)
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• Length fluctuations
• operators (states of the spin chain) of
  different length mix
• Hamiltonian is a part of non-abelian symmetry
  group
• conformal group SO(4,2)SU(2,2) is part of
  PSU(2,24)
• so(4,2) Mµ? - rotations
• Pµ - translations
• Kµ - special conformal
  transformations
• D - dilatation

Ground state tr ZZZZ breaks PSU(2,24) ?
P(SU(22)xSU(22))
Bootstrap SU(22)xSU(22) invariant S-matrix

asymptotic
Bethe ansatz spectrum of an
infinite spin chain
Beisert05
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Beisert,Staudacher05
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STRINGS
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String theory in AdS5?S5
Metsaev,Tseytlin98
constant RR 4-form flux

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

Classical limit is
Bena,Polchinski,Roiban03
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AdS sigma-models as supercoset
S5 SU(4)/SO(5)
AdS5 SU(2,2)/SO(4,1)
AdS superspace
Super(AdS5xS5) PSU(2,24)/SO(5)xSO(4,1)
Z4 grading
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Coset representative g(s)
Currents j g-1dg j0 j1 j2 j3
Action
Metsaev,Tseytlin98
In flat space
Green,Schwarz84
no kinetic term for fermions!
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Degrees of freedom
Bosons 15 (dim. of SU(2,2)) 15 (dim. of
SU(4)) - 10 (dim. of SO(4,1)) -
10 (dim. of SO(5)) 10 (5 in
AdS5 5 in S5) - 2
(reparameterizations) 8
Fermions - bifundamentals
of su(2,2) x su(4) 4 x 4 x 2
32 real components
2 kappa-symmetry
2 (eqs. of motion are first order)
8
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Quantization
Berenstein,Maldacena,Nastase02 Callan,Lee,McLough
lin,Schwarz, Swanson,Wu03 Frolov,Plefka,Zamaklar
06

• fix light-cone gauge and quantize
• action is VERY complicated
• perturbation theory for the spectrum, S-matrix,
• study classical equations of motion (gauge
  unfixed), then guess
• quantize near classical string solutions

Callan,Lee,McLoughlin,Schwarz,Swanson,Wu03
Klose,McLoughlin,Roiban,Z.in progress
Kazakov,Marshakov,Minahan,Z.04
Beisert,Kazakov,Sakai,Z.05 Arutyunov,Frolov,Sta
udacher04 Beisert,Staudacher05
Frolov,Tseytlin03-04 Schäfer-Nameki,Zamaklar,Z.
05 Beisert,Tseytlin05 Hernandez,Lopez06
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Consistent truncation
String on S3 x R1
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Gauge condition
Equations of motion
Zero-curvature representation
equivalent
Zakharov,Mikhaikov78
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Classical string Bethe equation
Kazakov,Marshakov,Minahan,Z.04
Normalization
Momentum condition
Anomalous dimension
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Quantum string Bethe equations
Arutyunov,Frolov,Staudacher04
extra phase
Beisert,Staudacher05
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Arutyunov,Frolov,Staudacher04
Hernandez,Lopez06

• Algebraic structure is fixed by symmetries
• The Bethe equations are asymptotic they describe
  infinitely long strings / spin chains and do not
  capture finite-size effects.

Beisert05
Schäfer-Nameki,Zamaklar,Z.06
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Open problems

• Interpolation from weak to strong coupling in the
  dressing phase
• How accurate is the asymptotic BA? (Probably up
  to
• e-f(?)L)
• Eventually want to know closed string/periodic
  chain spectrum
• need to understand
  finite-size effects
• Algebraic structure
• Algebraic Bethe ansatz?
• Yangian symmetries?
• Baxter equation?

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