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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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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
sl(2) sector:
Can be diagonalized by BAE
In scaling limit the Bethe roots condense into cuts
Cuts of roots correspond to the classical solutions
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
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
BAE for SU(1,2) spin chaine
Where
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
Expanding in L we get Where
and
On C23
On C13
Taking into account this mismatch we can write equation for density
Bethe roots form bound states, but they are separated by 1
Beisert, Staudacher;Beisert,Eden,Staudacher
For general configuration of roots we have the following equation
Where
From “stack” to “zipper”
Bosonic duality