two scalar fields of the n=4 sym theory: long local operators: can be mapped to the spin chain...
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Two scalar fields of the N=4 SYM theory: Long local operators: Can be mapped to the spin chain states: The mixing matrix is an integrable spin chain Hamiltonian! Minahan, ZaremboTRANSCRIPT
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Two scalar fields of the N=4 SYM theory:
Long local operators:
Can be mapped to the spin chain states:
The mixing matrix is an integrable spin chain Hamiltonian!Minahan, Zarembo
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sl(2) sector:
Can be diagonalized by BAE
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In scaling limit the Bethe roots condense into cuts
Cuts of roots correspond to the classical solutions
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Expanding Bathe ansatz equation for sl(2) spin chain we will find Korchemsky; Kazakov; Beisert, Tseytlin, Zarembo
where
Then the BAE becomes to the 1/L order
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Korchemsky; N.G. V Kazakov
BAE is equivalent to the absence of poles at u=uj
Baxter “polynomial”
Let us define q(x) by the following equation
exp(i q(x)) is a double valued function
Expanding T(u) We get for q
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BAE for SU(1,2) spin chaine
Where
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N.G. P. VieiraFor su(2,1) spin chain there are several Baxter polynomials
We can define some algebraic curve by the polynomial equation
Then for each branch cut we must have
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Expanding in L we get Where
and
On C23
On C13
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Taking into account this mismatch we can write equation for density
Bethe roots form bound states, but they are separated by 1
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Beisert, Staudacher;Beisert,Eden,Staudacher
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For general configuration of roots we have the following equation
Where
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From “stack” to “zipper”
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Bosonic duality