Transcript Bethe Ansatz in AdS/CFT: from local operators to classical

Recent developments
in AdS/CFT correspondence
Kostya Zarembo
École Normale Supérieure
&
Uppsala U.
Rencontres de Moriond, La Thuile, 10.03.2008
Yang-Mills fields = closed strings
or maybe
≈
“rings of glue”
Polyakov’80
Planar diagrams and strings
Large-N limit:
‘t Hooft’74
time
Conceptual problems:
• Closed strings describe gravity. What is graviton
in YM?
• String theory is only consistent in ten dimensions.
• How does the string remember that it is made of
gluons?
String states
Bound states in QFT
(mesons, glueballs)
String states
Local operators
Resolves many puzzles of putative large-N string:
• graviton is not a massless glueball, but is the dual of
the energy-momentum tensor Tμν
• extra dimensions are geometric images of the energy
scale and of global symmetries
• sum rules are automatic
AdS/CFT correspondence
Yang-Mills theory with
N=4 supersymmetry
Exact equivalence
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
String theory on
AdS5xS5 background
Anti-de-Sitter space (AdS5)
z
5D bulk
0
4D boundary
AdS/CFT correspondence
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Strong-weak coupling interpolation
λ
0
SYM perturbation
theory
1+
+
String perturbation
theory
+…
Circular Wilson loop (exact):
Erickson,Semenoff,Zarembo’00
Drukker,Gross’00
Pestun’07
Minimal area law in AdS5
Correlation functions
Dilatation operator:
matrix of anomalous dimensions
Local operators and spin chains
i
j
i
j
One loop planar mixing matrix:
Minahan,Z.’02
Heisenberg Hamiltonian
The spectrum
Ground state:
Excited states (magnons):
scattering phase shifts
momentum
Exact periodicity condition:
periodicity of wave function
Bethe equations for Heisenberg model
Rapidity:
Bethe’31
Anomalous dimension:
Full asymptotic BA
Beisert,Staudacher’05
Wrapping/finite size effects
Ambjørn,Janik,Kristjansen’05
Weak coupling:
captured by ABA
wrapping order
Beisert,Kristjansen,Staudacher’03
Large-spin twist-2 operators
where
Beisert,Eden,Staudacher’06
Weak coupling:
Beisert,Eden,Staudacher’06
Confirmed by explicit four-loop calculations in SYM
Bern,Czakon,Dixon,Kosower,Smirnov’06; Cachazo,Spradlin,Volovich’06
Strong coupling:
Basso,Korchemsky,Kotański’07
Confirmed by explicit two-loop calculations in
string theory
Roiban,Tseytlin’07; Klose,Minahan,McLoughlin,Z.’07
Conclusions
• Large-N SYM is a solvable (and to certain degree
solved) 4d theory. The solution is not simple!
• Finite-size/wrapping effects (Asymptotic Bethe
Ansatz => Thermodynamic Bethe Ansatz?)
• String theory <=> Yang-Mills <=> spin chain
How does it work in detail?
• What can we learn from solving AdS/CFT exactly?