Several recent works [1][2][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 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 2 1 ≤ . . . ≤ f 2 N 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 < √ N M pl . We then estimate the actions of instantons that fulfill the WGC. The leading instanton action is bounded from below by S ≥ SM pl /f N , with S a fixed constant, but in the universal limit S S √ 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.

We argue that super-Planckian diameters of axion fundamental domains can arise in Calabi-Yau compactifications of string theory. In a theory with N axions θ i , the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form −π < Q i j θ j < π. We compute the diameter of the fundamental domain in terms of the eigenvalues f 2 1 ≤ . . . ≤ f 2 N of the metric on field space, and also, crucially, the largest eigenvalue of (QQ ) −1 . At large N , QQ approaches a Wishart matrix, due to universality, and we show that the diameter is at least N f N , exceeding the naive Pythagorean range by a factor > √ N . This result is robust in the presence of P > N constraints, while for P = N the diameter is further enhanced by eigenvector delocalization to N 3/2 f N . We directly verify our results in explicit Calabi-Yau compactifications of type IIB string theory. In the classic example with h 1,1 = 51 where parametrically controlled moduli stabilization was demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys f N ≈ 0.013M pl . The random matrix analysis then predicts, and we exhibit, axion diameters ≈ M pl for the precise vacuum parameters found in [1]. Our results provide a framework for pursuing large-field axion inflation in well-understood flux vacua.

Abstract:We study the distribution of metastable vacua and the likelihood of slow roll inflation in high dimensional random landscapes. We consider two examples of landscapes: a Gaussian random potential and an effective supergravity potential defined via a Gaussian random superpotential and a trivial Kähler potential. To examine these landscapes we introduce a random matrix model that describes the correlations between various derivatives and we propose an efficient algorithm that allows for a numerical study of high dimensional random fields. Using these novel tools, we find that the vast majority of metastable critical points in N dimensional random supergravities are either approximately supersymmetric with |F | M susy or supersymmetric. Such approximately supersymmetric points are dynamical attractors in the landscape and the probability that a randomly chosen critical point is metastable scales as log(P ) ∝ −N . We argue that random supergravities lead to potentially interesting inflationary dynamics.

We determine the spectrum of scalar masses in a supersymmetric vacuum of a general N = 1 supergravity theory, with the Kähler potential and superpotential taken to be random functions of N complex scalar fields. We derive a random matrix model for the Hessian matrix and compute the eigenvalue spectrum. Tachyons consistent with the Breitenlohner-Freedman bound are generically present, and although these tachyons cannot destabilize the supersymmetric vacuum, they do influence the likelihood of the existence of an 'uplift' to a metastable vacuum with positive cosmological constant. We show that the probability that a supersymmetric AdS vacuum has no tachyons is formally equivalent to the probability of a large fluctuation of the smallest eigenvalue of a certain real Wishart matrix. For normally-distributed matrix entries and any N , this probability is given exactly by P = exp(−2N 2 |W | 2 /m 2 susy ), with W denoting the superpotential and m susy the supersymmetric mass scale; for more general distributions of the entries, our result is accurate when N 1. We conclude that for |W | m susy /N , tachyonic instabilities are ubiquitous in configurations obtained by uplifting supersymmetric vacua.

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