Several recent works 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 $N\gg1$ axions, super-Planckian axion diameters $\cal{ D}$ are readily allowed by the WGC. We clarify the nontrivial relationship between the kinetic matrix $K$ --- unambiguously defined 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 $f_1^2 \le ... \le f_N^2$ 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 ${ \cal D}<M_{ pl}$ nor ${ \cal D}<\sqrt{ N}M_{ pl}$. We then estimate the actions of instantons that fulfill the WGC. The leading instanton action is bounded from below by $S \ge { \cal S} M_{ pl}/f_N$, with ${ \cal S}$ a fixed constant, but in the universal limit $S\gtrsim { \cal S} \sqrt{ N}M_{ pl}/f_N$. 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.
Categories: hep-th