Mini-Workshop: Singularities in ${\mathrm{G}}_2$-geometry

Abstract

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.