Integrable systems, separation of variables and the Yang-Baxter equation

Abstract

This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian Y(gl(n)) of gl(n) in particular the Bethe algebra, fusion and T-and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for high-rank gl(n) bosonic spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous part, which was operatorial, this approach is functional and is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains.The final part of this work is based on the development of tools for solving the Yang-Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 × 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.

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

“…• Secondly, the FSoV approach, which we use and extend here, does not require the existence of a highest-weight state. As such our approach is applicable to models which do not have the highest-weight state, for example the conformal spin chain (fishchain) 12 [42] describing correlators with non-trivial coupling dependence in 4D conformal fishnet theory.…”

“…Remarkably, certain three point correlation functions in N " 4 SYM have indeed been shown to take an incredibly simple form when expressed in terms of the QSC Q-functions [3][4][5] and the resulting expressions are reminiscent of correlation functions in integrable spin chains when expressed in separated variables. This observation has been one of the main driving factors in the development of SoV methods for higher rank [6][7][8][9][10][11] integrable spin chains which was, until recently, only applicable to the simplest rank one models with slp2q symmetry, see [12] for a recent comprehensive review.…”

Integrability structures in string theory

“…For more details see e.g. [45,46]. Suppose, each particle in addition to rapidities is described by a linear space V of its states, then R-matrices R ij ∈ End(V i ⊗ V j ) act on the vector space of states of two particles encoding their interaction.…”

Minuscule ABCDE Lax operators from 4D Chern-Simons theory

“…Few years ago, a four dimensional (4D) topological gauge theory with complexified gauge symmetry G [1,2] has been proposed to be a mother theory of lower dimensional integrable systems such as quantum 1D integrable spin chains of statistical mechanics [4]- [5] and 2D integrable QFTs [6]- [8]. This 4D topological gauge theory is a tricky extension of the usual non abelian 3D Chern-Simons theory [3] with observables given by topological line defects such as the electrically charged Wilson lines and the magnetically charged 't Hooft lines.…”