Mini-Workshop: Singularities in $\mathrm G_2$-geometry
Anda Degeratu, Mark Haskins, Hartmut Weiß · Oberwolfach Reports · 2015
All currently known construction methods of smooth compact \mathrm G_2 -manifolds have been tied to certain singular \mathrm G_2 -spaces, which in Joyce’s original construction are \mathrm G_2 -orbifolds and in Kovalev’s twisted connected sum construction are complete G2-manifolds with cylindrical ends. By a slight abuse of terminology we also refer to the latter as singular \mathrm G_2 -spaces, and in fact both construction methods may be viewed as desingularization procedures. In turn, singular \mathrm G_2 -spaces comprise a (conjecturally large) part of the boundary of the moduli space of smooth compact \mathrm G_2 -manifolds, and so their deformation theory is of considerable interest. Furthermore, singular \mathrm G_2 -spaces are also important in theoretical physics. Namely, in order to have realistic low-energy physics in M-theory, one needs compact singular \mathrm G_2 -spaces with both codimension 4 and 7 singularities according to Acharya and Witten. However, the existence of such singular \mathrm G_2 -spaces is unknown at present. The aim of this workshop was to bring reserachers from special holonomy geometry, geometric analysis and theoretical physics together to exchange ideas on these questions.