The asymptotic geometry of G$_2$-monopoles

Início: 15/10/2020 17:00

Término: 15/10/2020 18:00

Palestrante: Daniel Fadel (Universidade Federal Fluminense)

E-mail do Palestrante:

Resumo: G$_2$-geometry is a very rich and vast subject in Differential Geometry which has been seeing a lot of progress in the last two decades. There are by now very powerful methods that produce millions of examples of G$_2$ holonomy metrics on the compact setting and infinitely many on the non-compact setting. Besides these fruitful advances, at present, there is no systematic understanding of these metrics. In fact, a very important problem in G$_2$-geometry is to develop methods to distinguish G$_2$-manifolds. One approach intended at producing invariants of G$_2$-manifolds is by means of higher dimensional gauge theory. G$_2$-monopoles are solutions to a first order nonlinear PDE for pairs consisting of a connection on a principal bundle over a noncompact G$_2$-manifold and a section of the associated adjoint bundle. They arise as the dimensional reduction of the higher dimensional Spin$(7)$-instanton equation, and are special critical points of an intermediate energy functional related to the Yang-Mills-Higgs energy. Donaldson-Segal (2009) suggested that one possible approach to produce an enumerative invariant of (noncompact) G$_2$-manifolds is by considering a ``count" of G$_2$-monopoles and this should be related to conjectural invariants ``counting" rigid coassociate (codimension 3 and calibrated) cycles. Oliveira (2014) started the study of G$_2$-monopoles providing the first concrete non-trivial examples and giving evidence supporting the Donaldson-Segal program by finding families of G$_2$-monopoles parametrized by a positive real number, called the mass, which in the limit when such parameter goes to infinity concentrate along a compact coassociative submanifold. In this talk I will explain some recent results, obtained in collaboration with Ákos Nagy and Gonçalo Oliveira, which show that the asymptotic behavior satisfied by the examples are in fact general phenomena which follows from natural assumptions such as the finiteness of the intermediate energy. This is a very much needed development in order to produce a satisfactory moduli theory and making progress towards a rigorous definition of the putative invariant. Time permitting, I will mention some interesting open problems and possible future directions in this theory.