Curved Yang-Mills gauge theories and their recent applications

par Simon-Raphael Fischer (National Center for Theoretical Sciences, National Taiwan University, Taiwan)

vendredi 16 févr. 2024, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Bâtiment Braconnier)

We will study curved Yang-Mills gauge theories, motivated by the infinitesimal gauge theory developed by Alexei Kotov and Thomas Strobl. These gauge theories are based on principal bundles equipped with a Lie group bundle action, in contrast to a Lie group action. In order to define a gauge invariant theory we have to introduce a connection on the structural Lie group bundle, and we impose conditions on its connection 1-form: It has to be a multiplicative form, i.e. closed w.r.t. a certain simplicial differential, and its curvature has to be exact; the connection 1-form is a generalization of the Maurer-Cartan form of the classical gauge theory, while the exactness of its curvature will generalize the role of the Maurer-Cartan equation. For allowing curved connections, while preserving gauge invariance, we will need to generalize the typical definition of the curvature/field strength on the principal bundle, by essentially accounting the curvature on the group bundle. If the connection on the group bundle is flat, the gauge theory is called to be flat and thus (almost) an ordinary gauge theory.

Recently, it was shown that such gauge theories help to classify singular foliations which we will discuss afterwards; the previously-mentioned connection 1-form is then what one calls a connection preserving the foliation, encoding a sort of horizontal part of a singular foliation. In return, the theory of singular foliations will help us to classify whether a curved gauge theory is equivalent to a flat gauge theory.