integrability and bethe ansatz in the ads/cft correspondence konstantin zarembo (uppsala u.) nordic...
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Integrability and Bethe Ansatz in the AdS/CFT correspondence
Konstantin Zarembo
(Uppsala U.)
Nordic Network MeetingHelsinki, 27.10.05
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)
Large-N expansion of gauge theory
String theory
Early examples:
• 2d QCD
• Matrix models
4d gauge/string duality:
• AdS/CFT correspondence
‘t Hooft’74
Brezin,Itzykson,Parisi,Zuber’78
Maldacena’97
Plan
1. Large-N limit and planar diagrams
2. Instead of an introduction: local operators=closed string states
3. Operator mixing and intergable spin chains
4. Basics of Bethe ansatz
5. Thermodynamic limit
I. GAUGE THEORY
II. STRING THEORY
1. Classical integrability
2. Classical Bethe ansatz
3. (time permitting) Quantum corrections
Yang-Mills theory
anti-Hermitean traceless NxN matrices
Interesting case: N=3 But we keep N as a parameter
Large-N limit‘t Hooft’74
“Index conservation law”:
Planar diagrams and strings
time
‘t Hooft coupling:
String coupling constant =
(kept finite)
(goes to zero)
AdS/CFT correspondence Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Anti-de-Sitter space (AdS5)
5D bulk
4D boundary
z
0
z
0
string propagator
in the bulk
Two-point correlation functions
Scale invariance
leaves metric
invariant
dual gauge theory is scale invariant (conformal)
Breaking scale invariance
“IR wall”
UV boundary
asymptotically
AdS metric
approximate
scale invariance
at short distances
If there is a string dual of QCD, this resolves many
puzzles:
• graviton is not a massless glueball, but the dual of Tμν
• sum rules are automatic
String statesBound states in QFT
(mesons, glueballs)
String states Local operators
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.
“IR wall”
UV boundary
asymptotically
AdS
The simplest phenomenological model describes all data in the
vector meson channel to 4% accuracy
(Spectral representation of bulk-to-boundary propagator)
Erlich,Katz,Son,Stephanov’05
λ<<1 Quantum strings
Classical strings Strong coupling in SYM
Way out: consider states with large quantum numbers
= operators with large number of constituent fields
Macroscopic strings from planar diagrams
Large orders
of perturbation theoryLarge number
of constituentsor
Price: highly degenerate operator mixing
Operator mixing
Renormalized operators:
Mixing matrix (dilatation operator):
Multiplicatively renormalizable operators
with definite scaling dimension:
anomalous dimension
N=4 Supersymmetric Yang-Mills Theory
Field content:
The action:
Brink,Schwarz,Scherk’77
Gliozzi,Scherk,Olive’77
Local operators and spin chains
related by SU(2) R-symmetry subgroup
i j
i j
• ≈ 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
Operator basis:
One loop planar (N→∞) diagrams:
Permutation operator:
Integrable Hamiltonian! Remains such
• at higher orders in λ
• for all operators
Beisert,Kristjansen,Staudacher’03; Beisert’03; Beisert,Dippel,Staudacher’04
Beisert,Staudacher’03
Spectrum of Heisenberg ferromagnet
Excited states:
Ground state:
flips one spin:
(SUSY protected)
• good approximation if M<<L
Exact solution:
• exact eigenstates are still multi-magnon Fock states
• (**) stays the same
• only (*) changes!
Non-interacting magnons
Exact periodicity condition:
momentumscattering phase shifts
periodicity of wave function
Zero momentum (trace cyclicity) condition:
Anomalous dimension:
Bethe’31
Bethe ansatz
Rapidity:
How to solve Bethe equations?
Non-interactions magnons:
mode number
Thermodynamic limit (L→∞):
u
0
bound states of magnons – Bethe “strings”
mode numbers
u
0
Sutherland’95;
Beisert,Minahan,Staudacher,Z.’03
Macroscopic spin waves: long strings
defined on cuts Ck in the complex plane
Scaling limit:
x
0
In the scaling limit,
determines the branch of log
Taking the logarithm and expanding in 1/L:
Classical Bethe equations
Normalization:
Momentum condition:
Anomalous dimension: