We consider topology changing transitions for $M$-theory compactifications on Calabi-Yau fourfolds with background $G$-flux. The local geometry of the transition is generically a genus g curve of conifold singularities, which engineers a 3d gauge theory with four supercharges, near the intersection of Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta which solve the local quantization condition on $G$ for a given geometry, including new solutions in which the flux is neither of horizontal nor vertical type. A local analysis of the flux superpotential shows that the potential has flat directions for a subset of these fluxes and the topologically different phases can be dynamically connected. For special geometries and background configurations, the local transitions extend to extremal transitions between global fourfold compactifications with flux. By a circle decompactification the $M$-theory analysis identifies consistent flux configurations in four-dimensional $F$-theory compactifications and flat directions in the deformation space of branes with bundles.
Kenneth Intriligator. Hans Jockers. Peter Mayr. David R. Morrison. M. Ronen Plesser. "Conifold transitions in $M$-theory on Calabi-Yau fourfolds with background fluxes." Adv. Theor. Math. Phys. 17 (3) 601 - 699, June 2013.