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

Abstract

This thesis is based on the author’s publications during the course of his PhD studies and focuses on various aspects of the field of quantum integrable systems. The aim of this thesis is to develop the so-called separation of variables program for high rank integrable systems and to develop new efficient techniques to solve one of the central equations of integrability, the Yang-Baxter equation. It is divided into five parts. The first part is an overview of the subject of quantum integrability in particular its mathematical description in terms of quantum algebras. We review standard textbook material and explain how various objects of physical interest such as Hamiltonians and S-matrices fit into the picture. The second part reviews the state-of-the-art of the separation of variables (SoV) program and discusses the author’s own contributions of this area and is based on the author’s publications [1] and [5]. 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, in particular what we refer to as the embedding morphism. The third part is based on the author’s publication [3] and part of the publication [7]. We build on the previous part and develop new techniques for the computation of scalar products in the SoV framework. 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. The fourth part is also based on the publication [7] and generalises the results of the previous part from compact spin chains to non-compact spin chains and also contains unpublished work of the author relating to non-compact Gelfand-Tsetlin patterns and their relation to hypergeometric functions. We also extend the previously mentioned functional formalism for computing scalar products to more non-trivial quantities such as form-factors of various operators including certain local operators. The final part of this thesis is based on the development of tools for solving the Yang-Baxter equation. It is primarily based on the publications [2, 6] with some reference to the publication [4] and the preprint [8]. 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. A summary of the work and directions for future research are presented at the end as well as an appendix to supplement the main text.