The fate of unstable gauge flux compactifications

Fluxes are widely used to stabilise extra dimensions, but the supporting monopole-like configurations are often unstable, particularly if they arise as gauge flux within a non-abelian gauge sector. We here seek the endpoint geometries to which this instability leads, focussing on the simplest concrete examples: sphere-monopole compactifications in six dimensions. Without gravity most monopoles in non-abelian gauge groups are unstable, decaying into the unique stable monopole in the same topological class. We show that the same is true in Einstein-YM systems, with the new twist that the decay leads to a shrinkage in the size of the extra dimensions and curves the non-compact directions: in D dimensions a Mink(D-2) x S-2 geometry supported by an unstable monopole relaxes to AdS(D-2) x S-2, with the endpoint sphere smaller than the initial one. For supergravity the situation is more complicated because the dilaton obstructs such a simple evolution. The endpoint instead acquires a dilaton gradient, thereby breaking some of the spacetime symmetries. For 6D supergravity we argue that it is the 4D symmetries that break, and examine several candidates for the endpoint geometry. By using the trick of dimensional oxidation it is possible to recast the supergravity system as a higher-dimensional Einstein-YM monopole, allowing understanding of this system to guide us to the corresponding endpoint. The result is a Kasner-like geometry conformal to Mink(4) x S-2, with nontrivial conformal factor and dilaton breaking the maximal 4D symmetry and generating a singularity. Yet the resulting configuration has a lower potential energy than did the initial one, and is perturbatively stable, making it a sensible candidate endpoint for the evolution.