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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
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2 Plan
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Bohr-Sommerfield quantization Poles condensation
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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
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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]
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Excitations Read energy shift from the change in Position of the new pole is given by [Beisert, Freyhilt]
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9 Comparison: excitation energies We find: [Frolov, Tseytlin] [N.G. Pedro Vieira]
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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]
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Fluctuation energies - the n plane BMN frequencies Along the cuts we can drop the cot with exponential precision (in J) [Schafer-Nameki]
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Fluctuation energies - the n plane With given by Implies
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Fluctuation energies - the x plane FE corresponds to new pole atFor BMN solution
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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
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17 All-loop Bethe equations [Beisert, Staudacher]
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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
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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!
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Charges given by The Hernandez-Lopez conjectured coefficients! [Hernandez,Lopez; Freyhult, Kristansen] In terms of charges
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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
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22 [N.G. Pedro Vieira, (to appear)] [Korchemsky; Kazakov; Beisert, Tseytlin, Zarembo]
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Anomaly terms
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Thus we must add to the rhs of the classical equations The famous anomaly terms. Anomaly terms
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26 Relabeling Natural labeling: [G. Vieira]
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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
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30 Bootstrap approach SU(2) chiral model [A,B,Zamolodchikov, A.B.Zamolodchikov. 1977]
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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
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32 Nesting
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33 Conclutions
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