Axion Monodromy and the Weak Gravity Conjecture
  Axions with broken discrete shift symmetry (axion monodromy) have recently
played a central role both in the discussion of inflation and the `relaxion'
approach to the hierarchy problem. We suggest a very minimalist way to
constrain such models by the weak gravity conjecture for domain walls: While
the electric side of the conjecture is always satisfied if the
cosine-oscillations of the axion potential are sufficiently small, the magnetic
side imposes a cutoff, $\Lambda^3 \sim m f M_{pl}$, independent of the height
of these `wiggles'. We compare our approach with the recent related proposal by
Ibanez, Montero, Uranga and Valenzuela. We also discuss the non-trivial
question which version, if any, of the weak gravity conjecture for domain walls
should hold. In particular, we show that string compactifications with branes
of different dimensions wrapped on different cycles lead to a `geometric weak
gravity conjecture' relating volumes of cycles, norms of corresponding forms
and the volume of the compact space. Imposing this `geometric conjecture',
e.g.~on the basis of the more widely accepted weak gravity conjecture for
particles, provides at least some support for the (electric and magnetic)
conjecture for domain walls.
