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Details of Grant 

EPSRC Reference: EP/S020888/1
Title: EPSRC-SFI - Solving Spins and Strings
Principal Investigator: Torrielli, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Trinity College Dublin
Department: Mathematics
Organisation: University of Surrey
Scheme: Standard Research
Starts: 01 September 2019 Ends: 31 August 2022 Value (£): 390,084
EPSRC Research Topic Classifications:
Mathematical Physics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
28 Nov 2018 EPSRC Mathematical Sciences Prioritisation Panel November 2018 Announced
Summary on Grant Application Form
In recent years, stemming from the study of a particular class of integrable systems, new remarkable mathematical structures have been discovered. These exotic algebraic constructions extend the standard framework of quantum groups to situations where novel unexpected phenomena are seen to emerge. Integrable systems have the property that their evolution equations can be exactly solved via reduction to an auxiliary linear problem. When these systems are combined with Lie superalgebras - that is, Lie algebras for which there exists a notion of "even" (commuting) and "odd" (anti-commuting) generators - new exciting facts occur. This has been partly established through the work of the applicants. The so called "Hopf" algebra describing multiple (tensorial) products of these algebras, for instance, acquires non-trivial deformations, whose consequences have not yet been fully understood. Furthermore, the systems in question exhibit a symmetry-enhancement which is not manifest from the Hamiltonian formulation. This "secret" symmetry results in novel more complicated quantities being conserved during the time evolution. A complete mathematical formulation of these effects has yet to be developed, and it is believed to be crucial to understand potential implications for branches of mathematics such as algebra, geometry, the topology of knots and link-invariants, and integrable systems.

The aim of this research project is to understand such exotic structures, and use this new understanding to attack challenging problems at the interface between Mathematical Physics and these contiguous areas. One such problem is the so-called "non-ultralocality" of Poisson structures, governing the formulation of integrable systems in their semi-classical approximation. Non-ultralocality makes the algebraic interpretation of the solution to these systems dramatically more obscure, and it is a difficult problem which has challenged mathematicians for years. We believe that the key to significant progress in this direction is a rigorous understanding of the underlying exotic algebras. Any progress in this area will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally.

We plan to attack the problem by constructing a diverse set of "representations" which explicitly realise the action of these exotic algebras; especially important will be what we call the "massless" ones. These are special representations occurring when the parameters satisfy very particular relations, and have recently been found to play a crucial role in the associated spectral analysis. This will be combined with the development of new techniques to treat quantum superalgebras and the so-called Bethe ansatz. From this work, we plan to derive new results on quantum groups and apply them to the problem of non-ultralocality in integrable systems. The intradisciplinary character of the project, combining ideas and techniques from different areas of mathematics, will lead to new results across a broad range of topics, from group theory to geometry (Hamiltonian structures), topology (knot invariants, Grassmannian manifolds) and combinatorics (Bethe equations, Baxter operators and Yangians).

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Organisation Website: http://www.surrey.ac.uk