A missing link: Brane networks and the Cobordism Conjecture

The absence of global symmetries in a quantum gravity theory often requires the introduction of (new) symmetry-breaking defects, which appear as singular objects in the low-energy description. This has been formalized in the Cobordism Conjecture, which further relates the asymptotics of these defects to non-trivial deformation classes of the effective theory. In this work we investigate the symmetry-breaking defects for theories with a discrete symmetry $G$ encoded in the bordism groups $\Omega^{\xi}_2 (BG)$ and, in particular, its sub-class described in terms of the homology groups $H_2(BG;\mathbb{Z})$. Contrary to expectations we find that the defects are of codimension two rather than three. However, they do not appear isolated but participate in brane networks explaining the naive mismatch. While in special situations linking configurations of defects are sufficient, our strategy generically predicts the existence of junctions, thus expanding the predictive power of the Cobordism Conjecture. We demonstrate the viability of this approach in four-dimensional supergravity theories originating from string and M-theory with a discrete Heisenberg group acting on its axionic degrees of freedom.