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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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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

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]

Excitations

Read energy shift from the change in

Position of the new pole is given by

[Beisert, Freyhilt]

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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]

Fluctuation energies - the n plane

BMN frequencies

Along the cuts we can drop the cot with exponential precision (in J)

[Schafer-Nameki]

Fluctuation energies - the n plane

With given by

Implies

Implies

Fluctuation energies - the x planeFE corresponds to new pole at For BMN solution

Fluctuation energies - the x planeFE 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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All-loop Bethe equations[Beisert, Staudacher]

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

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!

Charges given by

The Hernandez-Lopez conjectured coefficients!

[Hernandez,Lopez; Freyhult, Kristansen]

In terms of charges

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[N.G. Pedro Vieira, (to appear)]

[Korchemsky; Kazakov; Beisert, Tseytlin, Zarembo]

Anomaly terms

Thus we must addto the rhs of the classical equations

The famous anomaly terms.

Anomaly terms

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Relabeling

Natural labeling: [G. Vieira]

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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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Bootstrap approach

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

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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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Nesting

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Conclutions