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, 27.10.05
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Large-N expansion of gauge theory
String theory
Early examples

• 2d QCD
• Matrix models

t Hooft74
Brezin,Itzykson,Parisi,Zuber78
4d gauge/string duality

• AdS/CFT correspondence

Maldacena97
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Plan
I. GAUGE THEORY

• Large-N limit and planar diagrams
• Instead of an introduction local
  operatorsclosed string states
• Operator mixing and intergable spin chains
• Basics of Bethe ansatz
• Thermodynamic limit

II. STRING THEORY

• Classical integrability
• Classical Bethe ansatz
• (time permitting) Quantum corrections

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Yang-Mills theory
anti-Hermitean traceless NxN matrices
But we keep N as a parameter
Interesting case N3
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Large-N limit
t Hooft74
Index conservation law
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Planar diagrams and strings
time
(kept finite)
t Hooft coupling String coupling constant
(goes to zero)
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AdS/CFT correspondence
Maldacena97
Gubser,Klebanov,Polyakov98 Witten98
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Anti-de-Sitter space (AdS5)
z
5D bulk
strings
0
gauge fields
4D boundary
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Two-point correlation functions
z
string propagator in the bulk
0
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Scale invariance
leaves metric invariant
dual gauge theory is scale invariant (conformal)
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Breaking scale invariance
IR wall
asymptotically AdS metric
UV boundary
approximate scale invariance at short distances
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Bound states in QFT (mesons, glueballs)
String states
Local operators
String states

• If there is a string dual of QCD, this resolves
  many
• puzzles
• graviton is not a massless glueball, but the dual
  of Tµ?
• sum rules are automatic

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Perturbation theory
Spectral representation
Hence the sum rule
If n are all string states with right quantum
numbers, the sum is likely to diverge because of
the Hagedorn spectrum.
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IR wall
asymptotically AdS
UV boundary
(Spectral representation of bulk-to-boundary
propagator)
The simplest phenomenological model describes all
data in the vector meson channel to 4 accuracy
Erlich,Katz,Son,Stephanov05
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Quantum strings
?ltlt1
Strong coupling in SYM
Classical strings
Way out consider states with large quantum
numbers operators with large number of
constituent fields
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Macroscopic strings from planar diagrams
Large orders of perturbation theory
Large number of constituents
or
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Price highly degenerate operator mixing
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Operator mixing
Renormalized operators
Mixing matrix (dilatation operator)
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Multiplicatively renormalizable operators with
definite scaling dimension
anomalous dimension
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N4 Supersymmetric Yang-Mills Theory
Brink,Schwarz,Scherk77 Gliozzi,Scherk,Olive77

• Field content

The action
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Local operators and spin chains
related by SU(2) R-symmetry subgroup
j
i
i
j
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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

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One loop planar (N?8) diagrams
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Permutation operator

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

Beisert,Kristjansen,Staudacher03 Beisert03
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
• only () changes!

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periodicity of wave function
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Bethe ansatz
Rapidity
Bethe31
Zero momentum (trace cyclicity) condition
Anomalous dimension
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How to solve Bethe equations?
Non-interactions magnons
mode number
Thermodynamic limit (L?8)
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0
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bound states of magnons Bethe strings
0
mode numbers
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Macroscopic spin waves long strings
Sutherland95 Beisert,Minahan,Staudacher,Z.03
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Scaling limit
defined on cuts Ck in the complex plane
0
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In the scaling limit,
Taking the logarithm and expanding in 1/L
determines the branch of log
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Classical Bethe equations
Normalization
Momentum condition
Anomalous dimension