The project aims
1. To understand the structure of Bethe equations in nonultralocal integrable models, and to investigate the corresponding integrable models and their underlying algebraic structures.
2. To better understand the algebraic structures underlying supergroup and supercoset sigma models, especially the role played by nonultralocality.
3. To understand boundary integrability and its underlying symmetries in supergroup and supercoset models, especially those associated with the AdS/CFT correspondence.
The crucial natural question to ask of a problem in mathematical physics is whether we can solve it. Since most such problems are couched in the language of differential equations, the question is therefore of whether we can integrate the equations, of whether the problem is 'integrable'. The generalized mathematics of integrability is that of the structures which make some problems solvable in a way that others are not. Perhaps surprisingly, it turns out that fundamental physics makes much use of integrable models. The mathematical analysis of the central issue in modern fundamental physics, the relationship between the gauge theories of particle physics and the much more speculative string theory (the 'gauge/string correspondence'), has proved to be full of integrable models in recent years - but whereas conventional integrability refers to time-evolution, the crucial integrability in gauge theory is with the evolution of the energy scale.
Some integrable models are 'ultralocal' - that is, very well-behaved in terms of the localization of their interactions. Most of the mathematical techniques of quantum integrability apply principally to such models. Less well-behaved, 'nonultralocal' (non-UL) models are harder to handle, but keep re-appearing and increasing in importance. Sometimes their problems can be accommodated in a rather ad-hoc way, but a systematic understanding of them is lacking. This project attacks the problem from a number of directions. Some involve attempting to understand better the algebraic structures underlying known non-UL models, with or without a boundary, especially those of, or generalized from, the gauge/string correspondence. However, a new departure is to take existing ideas for handling non-UL models in the abstract and attempt 'reverse-engineer' the associated non-UL integrable models.
Key findings included
1) that generalized twisted Yangians are the governing symmetry of D3, 5 and 7 branes in the worldsheet description of the AdS5/CFT4 gauge/string correspondence, with
2) the discovery of new classes of such coideal structures and their generalization to the quantum algebra of the deformed Hubbard model.
3) It was also shown how to bring the nonultralocailty of sigma models under control and apply this to AdS/CFT, while
4) new results were proved concerning representations of quantized afffine algebras using their q-characters.
5) A generalization of the Bethe ansatz has been discovered which yields new results in combinatorics, and
6) a hierachy of commuting charges has been construicted for the quantised Burgers equation.