Integrable   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$ (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$ (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

United Kingdom

Russia

Nikolay Gromov

Nicolò Primi

Paul Ryan

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

We consider a cusped Wilson line with J insertions of scalar fields in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and prove that in a certain limit the Feynman graphs are integrable to all loop orders. We identify the integrable system as a quantum fishchain with open boundary conditions. The existence of the boundary degrees of freedom results in the boundary reflection operator acting non-trivially on the physical space. We derive the Baxter equation for Q-functions and provide the quantisation condition for the spectrum. This allows us to find the non-perturbative spectrum numerically.