Integrability is a beautiful phenomenon usually confined to one- or two-dimensional systems. Recently it was established that higher-dimensional quantum field theories, such as certain supersymmetric Yang-Mills theories in four dimensions and Chern-Simons theories in three dimensions, can also be integrable, at least in their planar limit. The appearance of integrability, yet to be understood, seems to be related to the AdS/CFT mechanism, which connects these field theories to string theories embedded in generalised Anti-de Sitter spacetimes. The emerging structures, such as factorised S-matrices, Bethe equations, thermodynamic Bethe ansatz, quantum algebras, etc. bear many similarities with well-known solvable models (Heisenberg magnets, Hubbard models, Sine-Gordon models, two-dimensional sigma models), but seem to generalise and unify them in yet-to-be understood ways.
This project aims to develop and employ the full power of the theory of integrable quantum systems to new models of quantum many-body spin systems in string theory. These models have been placed at the international centre-stage of developments in string theory in a series of recent surprising and unexpected mathematical connections, which relate spectra of free strings in the AdS5/S5 curved background to the spectra of the Heisenberg type spin Hamiltonians with non-local interactions. Several student projects of various complexity are available at the honours and PhD levels.