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