Integrable theories possess a sufficient number of conserved quantities so that one can solve these theories completely, at least in principle. As such, they play an important role in understanding the mathematical structures of physically relevant theories.
Models may be integrable classically, or quantum mechanically, and various techniques exist that allow one to express the solution to these models in a remarkably compact form. These range from the Bethe ansatz, devised by Bethe in 1931 in order to solve the Heisenberg spin chain, to integral formulae and algebraic formulations, which can encompass infinite and finite-size problems.
A significant development in quantum integrability was the development of quantum inverse scattering methods. Often the hidden symmetries behind the exact solvability of these models organise themselves in infinite-dimensional quantum groups. Remarkable applications of quantum groups and integrable systems to the AdS/CFT duality were discovered by Minahan and Zarembo in 2002, who observed the emergence of integrable structures in the spectral problem of the maximally supersymmetric Yang—Mills theory in four dimensions.
Quantum groups, Hopf algebras, and their representation theory provide an ideal framework for dealing with quantum integrable systems. The procedure of the so-called algebraic Bethe ansatz, which originates from the simultaneous diagonalisation of the quantum Hamiltonian and the higher commuting charges by means of creation and annihilation operators, can be reformulated in terms of the representation theory of a particular Hopf algebra based on the algebra known as RTT. This in turn, in the case of rational spin chains, leads directly to the theory of Yangians. The mathematics of Yangians and their deformations has had a vast impact in supersymmetric Yang—Mills theory, both in the study of the spectral problem and of amplitudes and correlation functions. We develop and analyse all these structures, and there are now many exciting new directions for integrability and quantum group theory with exciting cross-disciplinary aspects, e.g. in twistor geometry.