1. Gauge/String Duality and Integrable Systems
Kostya Zarembo
(Uppsala U.)
Annual Theory Meeting, Durham,

2. Yang-Mills fields = closed strings
“rings of glue”
or maybe ≈
Polyakov’80

3. string carries constant
Coulomb field vs Flux tube
(QED) (QCD) q q
string carries constant
energy per unit length

4. More refined lattice simulations confirm that the string fluctuates.
From Bali, Phys. Rep. 343 (2001) 1
More refined lattice simulations confirm that the string fluctuates.
Caselle,Fiore,Gliozzi,Hasenbusch,Provero’97; Caselle,Pepe,Rago’04; …

5. “Index conservation law”:
Large-N expansion
‘t Hooft’74
“Index conservation law”:

6. N N N N
Two expansion parameters:
‘t Hooft coupling:
string coupling:

7. Planar diagrams and strings
Large-N limit:
time

8. 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?

9. Large-N expansion of YM theory
String theory
Early examples:
2d QCD
Matrix models
‘t Hooft’74
Brezin,Itzykson,Parisi,Zuber’78
4d gauge/string duality:
AdS/CFT correspondence
Maldacena’97

10. Bound states in QFT
(mesons, glueballs)
String states
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

11. AdS/CFT correspondence
Yang-Mills theory with
N=4 supersymmetry
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98
Exact equivalence
String theory on
AdS5xS5 background

12. Anti-de-Sitter space (AdS5)z
5D bulk
strings
gauge fields
4D boundary

13. N=4 Supersymmetric Yang-Mills Theory
Gliozzi,Scherk,Olive’77
Brink,Schwarz,Scherk’77
Field content:
Action:
conformal field theory

14. AdS/CFT correspondence
Maldacena’97
Gubser,Klebanov,Polyakov’98
Witten’98

15. Strong-weak coupling interpolationλ
SYM perturbation
theory
String perturbation
theory 1 + +
+ …
Circular Wilson loop (exact):
Erickson,Semenoff,Zarembo’00
Drukker,Gross’00
Minimal area law in AdS5

17. Correlation functions
Dilatation operator:
matrix of anomalous dimensions

18. Operators Protected Non-degenerate Degenerate (mixes with )
energy-momentum tensor
Konishi operator
(mixes with )

19. Local operators and spin chainsj i j

20. Tree level: Δ=L (huge degeneracy)
One loop:

21. One loop planar dilatation generator:
Minahan,Z.’02
Heisenberg Hamiltonian
Integrability!

22. The spectrum
Ground state:
Excited states (magnons):
BMN (Berenstein,Maldacena,Natase’02) operators

25. Spectrum and scattering phase shifts
periodic short-range potential

26. exact only for V(x) = g δ(x)

27. Continuity of periodized wave function

28. where
is (eigenvalue of) the S-matrix
correct up to O(e-L/R)
works even for bound states via analytic continuation to complex momenta

29. Multy-particle states

30. Assumptions: Bethe equations R<<L
particles can only exchange momenta
no inelastic processes
integrability!

31. Exact periodicity condition:
momentum
scattering phase shifts
periodicity of wave function
Exact periodicity condition:

32. Bethe equations for Heisenberg model
Rapidity:
Bethe’31
Anomalous dimension:

33. Sigma-model in AdS5xS5 AdS5 S5 Classical limit
Metsaev,Tseytlin’98
AdS5 S5
Sigma-model coupling constant:
Classical limit is

34. φ
yi’ zi t S5
AdS5

35. Light-like geodesics:
Gauge fixing
Light-like geodesics:
gauge condition:
Berenstein,Maldacena,Nastase’02
Parnachev,Ryzhov’02
Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03
Arutyunov,Frolov,Plefka,Zamaklar’06

36. Bosonic Lagrangian in AdS5xS5 (up to quartic order in fields):
Massive, integrable 2d field theory
Lorentz-invariant kinetic terms
Lorentz invariance is broken by interactions
Gauge-dependent: a=1/2 (pure l.c. gauge) is the simplest case

37. Full asymptotic BA
Beisert,Staudacher’05

38. Wrapping/finite size effects
Ambjørn,Janik,Kristjansen’05
Weak coupling:
wrapping order
captured by ABA
Beisert,Kristjansen,Staudacher’03

39. since length of the string = and mass gap = 1 in light-cone gauge
Strong coupling:
Schäfer-Nameki,Zamaklar,Z.’06
since length of the string = and mass gap = 1
in light-cone gauge
asymptotic
exact

40. Large-spin twist-2 operators
where
Beisert,Eden,Staudacher’06

41. Weak coupling: Strong 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

42. 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?)
World-sheet sigma-model (continuous field theory) <=> spin chain (discrete system). How does it work? (Giant magnons? Hofman, Maldecena’06; Arutyunov,Frolov,Zamaklar’06)
What can we learn from solving AdS/CFT exactly?