Planckian Axions and the Weak Gravity Conjecture

TL;DR

The paper investigates whether the Weak Gravity Conjecture forbids super-Planckian axion field displacements in theories with many axions, without relying on monodromy. It shows that the axion diameter ${\cal D}$ cannot be inferred from the eigenvalues of the kinetic matrix alone; the orientation of eigenvectors relative to instanton identifications is crucial, and even $f_N\!<\!M_{pl}$ does not guarantee ${\cal D}<M_{pl}$. Gravitational instantons are shown to satisfy the zero-form WGC via a convex-hull condition, and in generic large-$N$ theories the minimal instanton action scales as $S_{\min} \gtrsim {\cal S}\sqrt{N}\,M_{pl}/f_N$, suppressing higher harmonics. Using Minkowski bounds and random-matrix ensemble analyses, the authors argue that the dominant instanton is controlled by the geometric mean of the eigenvalues rather than the largest, allowing large-field inflation to remain viable in many-axion setups.

Abstract

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.

Figures (2)

• Figure 1: The unit ball, contained in the convex hull of the charge-to-action vectors of four leading instantons (red) with action prefactor $S_{\text{leading}}$ and eight subleading instantons (blue) with action prefactor $S_{\text{subleading}}$.
• Figure 2: Left: Histogram of $\text{Min}_i({\mathscr Q}^i_R)/\rho_{\text{B}}$ for $N=2$ (faintest, rightmost peak) through $N=102$ (boldest, leftmost peak), for the Wishart ensemble. Right: $\langle \text{Min}_i({\mathscr Q}^i_R)/\rho_{\text{B}}\rangle$ over $N$. Evidently, typical shortest lattice vectors are roughly half the maximum length allowed by the Minkowski bound.