Transcript ppt/part 1

Integrability and Bethe Ansatz in the
AdS/CFT correspondence
Thanks to:
Konstantin Zarembo
(Uppsala U.)
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
Large-N expansion of gauge theory
String theory
Early examples:
• 2d QCD ‘t Hooft’74
• Matrix models Brezin,Itzykson,Parisi,Zuber’78
4d gauge/string duality:
• AdS/CFT correspondence
Maldacena’97
Plan
I. GAUGE THEORY
1.
2.
3.
4.
5.
Large-N limit and planar diagrams
Instead of an introduction: local operators=closed string states
Operator mixing and intergable spin chains
Basics of Bethe ansatz
Thermodynamic limit
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
“Index conservation law”:
‘t Hooft’74
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)
z
5D bulk
0
4D boundary
Two-point correlation functions
z
string propagator
in the bulk
0
Scale invariance
leaves metric
invariant
dual gauge theory is scale invariant (conformal)
Breaking scale invariance
“IR wall”
asymptotically
AdS metric
UV boundary
approximate
scale invariance
at short distances
String states
Bound states in QFT
(mesons, glueballs)
String states
Local operators
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
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”
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,Stephanov’05
λ<<1
Classical strings
Quantum 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 theory
or
Large number
of constituents
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
Brink,Schwarz,Scherk’77
Gliozzi,Scherk,Olive’77
Field content:
The action:
Local operators and spin chains
related by SU(2) R-symmetry subgroup
i
j
i
j
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
One loop planar (N→∞) diagrams:
Permutation operator:
Integrable Hamiltonian! Remains such
• at higher orders in λ Beisert,Kristjansen,Staudacher’03; Beisert’03; Beisert,Dippel,Staudacher’04
• for all operators Beisert,Staudacher’03
Spectrum of Heisenberg ferromagnet
Ground state:
(SUSY protected)
Excited states:
flips one spin:
Non-interacting magnons
• good approximation if M<<L
Exact solution:
• exact eigenstates are still multi-magnon Fock states
• (**) stays the same
• only (*) changes!
scattering phase shifts
momentum
Exact periodicity condition:
periodicity of wave function
Bethe ansatz
Rapidity:
Bethe’31
Zero momentum (trace cyclicity) condition:
Anomalous dimension:
How to solve Bethe equations?
Non-interactions magnons:
mode number
Thermodynamic limit (L→∞):
u
0
bound states of magnons – Bethe “strings”
u
0
mode numbers
Macroscopic spin waves: long strings
Sutherland’95;
Beisert,Minahan,Staudacher,Z.’03
Scaling limit:
defined on cuts Ck in the complex plane
x
0
In the scaling limit,
Taking the logarithm and expanding in 1/L:
determines the branch of log
Classical Bethe equations
Normalization:
Momentum condition:
Anomalous dimension: