Almost Duality on the small and large space
Lead Research Organisation:
University of Glasgow
Department Name: School of Mathematics & Statistics
Abstract
The Research project will address fundamental questions in the area Frobenius Manifolds and their applications. In the first part of the project, the algebraic and geometric properties of so-called V-systems will be studied, together with applications, such as those arising in the theory of bi-Hamiltonian geometry. These structures "live" on the small phase space. The second part of the project will apply these concepts to the infinite dimensional "big phase space", drawing on its dual interpretation as the arena of both Topological Quantum Field Theories and dispersive bi-Hamiltonian structures.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509668/1 | 01/10/2016 | 30/09/2021 | |||
| 1653916 | Studentship | EP/N509668/1 | 01/10/2015 | 27/12/2019 | Georgios Antoniou |
| Description | 1. Our research fills a gap in the investigation of certain supersymmetric quantum mechanical systems. Thus we were able to construct such supersymmetric extension of celebrated Calogero-Moser integrable system. This system describes pairwise interaction of particles on a line with inverse square distance potential. The extension which we discovered has four supercharges which are specific conserved quantities of the system. It was a long-standing problem to construct such a system. One of the motivations comes from considerations of black holes which have prescribed geometry. Furthermore, we also constructed versions of Calogero-Moser type supersymmetric systems corresponding to any V-system which is a special configuration of vectors in a space. We also extended our considerations to the case of Calogero-Moser-Sutherland system in which case one deals with a system of interacting particles on a circle rather than a line. 2. In the second part of the project we dealt with other Frobenius manifold structures associated with V-systems and finite groups generated by reflections. Here the geometrical object under consideration was a discriminant stratum which is a subspace of irregular orbits inside the space of orbits of the action of a finite reflection group on a vector space. We studied particular metric properties of these subspaces, thus we found nice compact expression for a volume form for each such subspace. These results are unexpected, they show that discriminant strata have nice and interesting metric properties. |
| Exploitation Route | 1. This area of research is very active and we believe the findings of part 1 can be useful to specialists with interests in theoretical physics as well as specialists in the integrable systems community. Thus we presented supersymmetric quantum models which in principle can be tested for description of entropy of black holes in accordance with conjecture of Gibbons and Townsend (from 1998). Also, future study of integrability of presented supersymmetric systems may potentially lead to new interesting structures. 2. Results on Coxeter discriminants may have applications in the theory of hyperplane arrangements. In particular our work produced a new class of multi-arrangements which are conjectured to be free by Douvropoulos and Stump (in 2019). Our results may also be of interest to many experts in Frobenius manifolds theory. |
| Sectors | Other |
| URL | http://theses.gla.ac.uk/id/eprint/79019 |