The topology of gauge groups and current groups for 3-manifolds

Lead Research Organisation: University of Southampton
Department Name: Sch of Mathematical Sciences

Abstract

Gauge groups and current groups are important in
mathematical physics, with gauge groups having important applications
in geometry and current groups having important applications in integrable
systems. Their topology is concerned with properties that remain unchanged
when continuously deformed. In both cases, the group can be described
via mapping spaces, which are better suited to analysis using methods
from algebraic topology. The mapping spaces involve a base space X and
a Lie group G. A great deal of recent work has been done investigating the
topology of gauge groups when X is a surface or a four-manifold, but not
a 3-manifold. Little has been done to investigate the topology of current
groups. This project aims to determine important properties of each in
the context of 3-manifolds, properties including their homology and
homotopy groups.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509747/1 01/10/2016 30/09/2021
1985336 Studentship EP/N509747/1 15/01/2018 15/01/2021 Simone Rea
 
Description This early stage of the research has focused on a generalisation of the arguments by Hasui, Kishimoto, Kono, and Sato on PU(3)-gauge groups over S^4 to the case of PU(n)-gauge groups over sphere of even dimension between 4 and 2n. Stronger conclusions are obtained when n is assumed to be prime. Crucially, it is shown that the order of the Samelson product of a generator of pi_3(PU(p)) with the identity on S^3 coincides with the order of the corresponding Samelson product for SU(p).

Special cases of our classification include known results such as the above mentioned result on PU(3)-gauge groups, as well as the result by Kamiyama, Kishimoto, Kono, Tsukuda on PU(2)-gauge groups over S^4.
Novel results include:
- a p-primary local classification of PU(5)-gauge groups over S^4;
- an integral classification of PU(3)-gauge groups over S^6.
Exploitation Route The results obtained thus far expand the list of principal bundles whose gauge groups are understood from the point of view of homotopy theory, and may prove relevant in a more global understanding of the problem. Researchers in theoretical physics areas involving gauge groups may also find these results useful.
Sectors Other