EPSRC Reference: 
EP/S020888/1 
Title: 
EPSRCSFI  Solving Spins and Strings 
Principal Investigator: 
Torrielli, Dr A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Surrey 
Scheme: 
Standard Research 
Starts: 
02 September 2019 
Ends: 
01 September 2022 
Value (£): 
390,084

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

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" (anticommuting) 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 nontrivial deformations, whose consequences have not yet been fully understood. Furthermore, the systems in question exhibit a symmetryenhancement 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 linkinvariants, 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 socalled "nonultralocality" of Poisson structures, governing the formulation of integrable systems in their semiclassical approximation. Nonultralocality 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 longterm 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 socalled Bethe ansatz. From this work, we plan to derive new results on quantum groups and apply them to the problem of nonultralocality 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 