The Mean Curvature of First-Order Submanifolds in Exceptional Geometries with Torsion

Abstract

We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G 2 -structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative 4-fold, Cayley 4-fold) is a minimal submanifold.In the process, we obtain new obstructions to the local existence of coassociative 4-folds in G 2 -structures with torsion.

New $G_2$-conifolds in $M$-theory and their Field Theory Interpretation

“…Advances have been made both from the physics [9,10,11,12,13,14] and the mathematics [15,1,16,17,18,19,20,21], and involves aspects such as a string-worldsheet interpretation [22,23,24], a better understanding of exceptional gauge bundles and instantons [25,26,27,28,29,30,31], in addition to non-perturbative aspects such as M 2 instantons and their relation to exceptional enumerative geometry [32,33,34]. On the physics side such new insights often rely on dualities with other string theories and F-theory [35,36,37,38], in addition to dimensional reductions [39,40,41,42,43].…”