1. 1 Integrability in AdS^5 x S_5 string theory: general 1-loop results Based on [N.G., Pedro Vieira] hep-th/0703191, hep-th/0703266, to appear

2. 2 Plan

3. 3

4. Bohr-Sommerfield quantization Poles condensation

5. 5

6. Integrability in AdS According to Beisert, Kazakov, Sakai and Zarembo, we can map a classical string motion to an 8-sheet Riemann surface Eigenvalues of a monodromy matrix

7. The electrostatic picture We can discretize the classical integral equations: Particles with the same n will condense into the same cut [Arutyunov, Frolov, Staudacher; Biesert, Staudacher]

8. Excitations Read energy shift from the change in Position of the new pole is given by [Beisert, Freyhilt]

9. 9 Comparison: excitation energies We find: [Frolov, Tseytlin] [N.G. Pedro Vieira]

10. 10

11. Fluctuation energies (FE) For the harmonic oscillator we have So far we understood how to get FE around any classical solution Now we can compute [Frolov, Tseytlin]

12. Fluctuation energies - the n plane BMN frequencies Along the cuts we can drop the cot with exponential precision (in J) [Schafer-Nameki]

13. Fluctuation energies - the n plane With given by Implies

14. Fluctuation energies - the x plane FE corresponds to new pole atFor BMN solution

15. Fluctuation energies - the x plane FE corresponds to new pole at For large n FE behave like but now we have new cuts We have precise split of two contributions

16. 16

17. 17 All-loop Bethe equations [Beisert, Staudacher]

18. Phase and Potential p’s are functions of the roots. In the large limit BAE’s without V(x) are We find the classical curve. With we have the same equations but

19. Deriving the Hernandez-Lopez phase If we add to the quasi-momenta the energy is shifted by the corresponding FE If we add all excitations we will find one loop shift. For example, for the first quasi-momentum we add The same we find for all quasi-momenta!

20. Charges given by The Hernandez-Lopez conjectured coefficients! [Hernandez,Lopez; Freyhult, Kristansen] In terms of charges

21. Review of the logic Extra poles in the classical algebraic curve Fluctuation energies Sum of all poles ij and integral over n = add V(x) to each quasi- momenta Derived HL phase Extra phase in BAE

22. 22 [N.G. Pedro Vieira, (to appear)] [Korchemsky; Kazakov; Beisert, Tseytlin, Zarembo]

23. Anomaly terms

24. Thus we must add to the rhs of the classical equations The famous anomaly terms. Anomaly terms

25. 25

26. 26 Relabeling Natural labeling: [G. Vieira]

27. 27

28. 28 Analytic properties of fluctuation energies Fluctuation energies are analytic functions of Fluctuation energies have branch cuts as functions of Branch cuts come from the map Since is a solution to Singularities are at

29. 29

30. 30 Bootstrap approach SU(2) chiral model [A,B,Zamolodchikov, A.B.Zamolodchikov. 1977]

31. 31 Derivation of the phase in Bootstrap approach In the large L, limit we can integrate out  ’s [NG, V.Kazakov] [G.Arutyunov, S.Frolov, M.Staudacher (hep-th/0406256)] For large and large number of particles density of rapidities is

32. 32 Nesting

33. 33 Conclutions