Planckian axions and the Weak Gravity Conjecture

Several recent works [1{3] have claimed that the Weak Gravity Conjecture (WGC) excludes super-Planckian displacements of axion fields, and hence large-field axion inflation, in the absence of monodromy. We argue that in theories with N >> 1 axions, super-Planckian axion diameters D are readily allowed by the WGC. We clarify the nontrivial relationship between the kinetic matrix K - unambiguously de fined by its form in a Minkowski-reduced basis - and the diameter of the axion fundamental domain, emphasizing that in general the diameter is not solely determined by the eigenvalues f(1)(2) <= . . . <= f(N)(2) of K: the orientations of the eigenvectors with respect to the identifications imposed by instantons must be incorporated. In particular, even if one were to impose the condition f(N) < M-pl, this would imply neither D < M-pl nor D < root NMpl. We then estimate the actions of instantons that fulfill the WGC. The leading instanton action is bounded from below by S >= SMpl = f(N), with S a fixed constant, but in the universal limit S greater than or similar to S root NMpl = f(N). Thus, having f(N) > M-pl does not immediately imply the existence of unsuppressed higher harmonic contributions to the potential. Finally, we argue that in effective axion-gravity theories, the zero-form version of the WGC can be satisfied by gravitational instantons that make negligible contributions to the potential.