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Recent developments in AdS/CFT correspondence

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 “rings of glue”. or maybe ≈. Polyakov’80. Planar diagrams and strings. Large-N limit:.

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Recent developments in AdS/CFT correspondence

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  1. Recent developments in AdS/CFT correspondence Kostya Zarembo École Normale Supérieure & Uppsala U. Rencontres de Moriond, La Thuile, 10.03.2008

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

  3. Planar diagrams and strings Large-N limit: ‘t Hooft’74 time

  4. 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?

  5. Bound states in QFT (mesons, glueballs) String states Local operators String states • 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

  6. 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

  7. Anti-de-Sitter space (AdS5) z 5D bulk strings 0 gauge fields 4D boundary

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

  9. Strong-weak coupling interpolation λ 0 SYM perturbation theory String perturbation theory 1 + + + … Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Pestun’07 Minimal area law in AdS5

  10. Correlation functions Dilatation operator: matrix of anomalous dimensions

  11. Local operators and spin chains j i j i

  12. One loop planar mixing matrix: Minahan,Z.’02 Heisenberg Hamiltonian

  13. The spectrum Ground state: Excited states (magnons):

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

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

  16. Full asymptotic BA Beisert,Staudacher’05

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

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

  19. 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

  20. 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? Conclusions

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