Moduli space of selfdual connections in dimension greater than four for abelian Gauge groups
Doctoral thesis
English
OPEN
Cappelle, Natacha
(2018)

Subject:
Moduli space  Gauge  YangMills  Selfdual connection
In 1954, C. Yang and R. Mills created a Gauge Theory for strong interaction of Elementary Particles. More generally, they proved that it is possible to define a Gauge Theory with an arbitrary compact Lie group as Gauge group. Within this context, it is interesting to find critical values of a functional defined on the space of connections: the YangMills functional. If the based manifold is four dimensional, there exists a natural notion of (anti)selfdual 2form, which gives a natural notion of (anti)selfdual connection. Such connections give critical values of the YangMills functional. Moreover, the Gauge group acts on the set of (anti)selfdual connections. The set of (anti)selfdual connections modulo the Gauge group is called the Moduli space of (anti)selfdual connections. It is interesting for physicists because it provides critical values of the YangMills functional and for mathematicians because it is an invariant of the based manifold. In dimension greater than four, it is possible to extend this notion of (anti)selfduality in different suitable ways. We are working on almost Kähler manifolds – so in particular on every symplectic manifolds. On almost Kähler manifolds, an extension of selfduality appears naturally. In Chapter 2, we define this extended notion of selfduality of 2forms and we determine the space of selfdual 2forms. In Chapter 3, we begin by explaining how this definition of selfduality of 2forms gives a definition of selfduality of connections. Then, we define the corresponding Moduli space of selfdual connections. Finally, we identify suitable hypotheses under which we are able to characterize the moduli space of selfdual connections and to build a Lie group structure on it.
(SC  Sciences)  UCL, 2018