Flux Compactifications Grow Lumps

The simplest flux compactifications are highly symmetric---a $q$-form flux is wrapped uniformly around an extra-dimensional $q$-sphere. In this paper, we investigate solutions that break the internal SO($q+1$) symmetry down to SO$(q)\times\mathbb Z_2$; we find a large number of such lumpy solutions, and show that often at least one of them has lower vacuum energy, larger entropy, and is more stable than the symmetric solution. We construct the phase diagram of lumpy solutions, and provide an interpretation in terms of an effective potential. Finally, we provide evidence that the perturbatively stable vacua have a non-perturbative instability to spontaneously sprout lumps; we give an estimate of the decay rate and argue that generically it is exponentially faster than all other known decays.