Planckian Axions in String Theory

Abstract

We argue that super-Planckian diameters of axion fundamental domains can naturally arise in Calabi-Yau compactifications of string theory. In a theory with $N$ axions $\hat{I}{¸}^i$, the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form $-\ddot{I}€<Q_j^i\hat{I}{¸}^j<\ddot{I}€$. We compute the diameter of the fundamental domain in terms of the eigenvalues $f_1^2\le \dot{.}.\le f_N^2$ of the metric on field space, and also, crucially, the largest eigenvalue of $(Q{Q}^{\top }{)}^{-1}$. At large $N$, $Q{Q}^{\top }$ 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 $>\sqrt{N}$. 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 $f_N\approx 0.013M_{pl}$. 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 achieving large-field axion inflation in well-understood flux vacua.

42 pages, 4 figures