Konishi:%20from%204%20loops%20to%20infinity

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Konishi from 4 loops to infinity
Nikolay Gromov Based on works with V.Kazakov,
P.Vieira A.Kozak
Symposium on Theoretical and Mathematical
Physics July 3-8, 2009
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AdS/CFT correspondence
AdS/CFT duality
String tension
t Hooft coupling
Anomalous dimensions spectrum of 2D integrable
field theories
Symmetry
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Integrability in N4 SYM
N4 Super Yang-Mills
- Mixing matrix integrable Hamiltonian
Minahan, Zarembo 20022008
The problem of diagonalization of the huge matrix
can be solved by Bethe ansatz
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Clasical integrability
Motion of the strin in AdS
Infinitely many Integrals of motion
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Integrability in 2D
Operator corresponding to an integral of motion
Between in and out states
The outgoing momenta are constrained
The only solution
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Factorization of S-matrix
Yang-Baxter equation
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Asymptotic Spectrum

• For spectral density we need finite volume
• From periodicity of the wave function-Bethe
  equations

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Bethe equations
Beisert, Staudacher Beisert, Hernandez,
Lopez Beisert,Eden,Staudacher Arutyunov, Frolov
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Vacuum
,Matsubara, Zamolodchikov,
I.e. from the asymptotical spectrum (infinite R)
we can compute the Ground state energy for ANY
finite volume!
In our case the wick rotated theory is different
mirror theory
Ambjorn, Janik, Kristjansen Arutynov, Frolov
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AdS/CFT bound states
Till Bargheer, Niklas Beisert, N. G.
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AdS/CFT Y-system
N.G., Kazakov, Vieira
Bazhanov, Lukyanov, Zamolodchikov, P.Dorey,
Totteo
Destri de Vega, Bytsko ,Teschner.
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Above ground state
Bazhanov, Lukyanov, Zamolodchikov, P.Dorey, Totteo
Inspiration 1 Analytical continuation
Inspiration 2 Integrable lattice models
Destri de Vega, Teschner
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Large L limit or weak coupling
N.G., Kazakov, Vieira
Use Hirota equation
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Hirota large L
The eigevalues solve Hirota! for rep
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Explicitly
Instead of computing the transfer matrices
explicitly one can use
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Konishi operator
The simplest operator
Bianchi, Kovacs,Rossi, Stanev Eden, Jarczak,
Sokatchev,Stanev
In agreement with perturbation theory!! 4-loops!
Kotikov, Lipatov, Rej, Staudacher and
Velizhanin Sieg, Torrielli Janik, Bojnok, N.G.,
Kazakov, Vieira
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Integral form of Y-system for exitations
Bombardelli, Fioravanti, Tateo N.G., Kazakov,
Vieira Arutynov, Frolov
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Exact BAE

• From mirror to physical

Precision is excellent!!!
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Konishi from weak to strong
Fit
From string side Roiban,Tseytlin
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Numerical results

• 20 computers 4 nights
• Emergence of integers a miracle
• Exact BAE modified by phase

Porto university Wolfram
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Comparison with Roiban and Tseytlin
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Conclutions

• Y-system is proven to be a computational tool
• We reproduced several leading terms at strong
  coupling for a simple operator
• Any sl2 operator could be computed readily for
  any g
• Generalization to other sectors to be done
• Analytical results coming soon?
• Other AdS/CFTs?

N.G., Kazakov, Vieira