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This article discusses how long local operators in N=4 SYM theory can be mapped to states in an integrable spin chain. The mixing matrix is an integrable spin chain Hamiltonian, and in the scaling limit, the Bethe roots condense into cuts that correspond to classical solutions. The article also explores the diagonalization of the sl(2) sector by BAE and the expansion of the Baxter ansatz equation for the sl(2) 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, 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
Korchemsky; Kazakov; Beisert, Tseytlin, Zarembo Expanding Bathe ansatz equation for sl(2) spin chain we will find where Then the BAE becomes to the 1/L order
Baxter “polynomial” Korchemsky; N.G. V Kazakov BAE is equivalent to the absence of poles at u=uj Let us define q(x) by the following equation exp(i q(x)) is a double valued function We get for q Expanding T(u)
For su(2,1) spin chain there are several Baxter polynomials N.G. P. Vieira 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
Bethe roots form bound states, but they are separated by 1 Taking into account this mismatch we can write equation for density
Beisert, Staudacher; Beisert,Eden,Staudacher
For general configuration of roots we have the following equation Where
