Flat Monodromies and a Moduli Space Size Conjecture

TL;DR

This work shows that type IIB flux on a toroidal orientifold can create flat, monodromic moduli spaces in which a two-axion system yields a long, winding axionic direction that is non-geodesic. The authors construct a superpotential $W=(M\tau_1-N\tau_2)(\tau-\tau_3)$ with a tadpole bound $MN\le16$, leading to an enlarged fundamental domain identified with a congruence subgroup $\Gamma^0(N)$ and a long but finite axionic trajectory. Crucially, the geodesic distance in moduli space grows only logarithmically with the flux $N$, and when a 4d cutoff $\Lambda$ is imposed, the diameter of the restricted moduli space scales like $\ln(1/\Lambda)$, forming a Moduli Space Size Conjecture. They also formulate a parallel bound along geodesics: there exist points where the lightest KK or winding mode mass is $\lesssim e^{-\alpha L}$ with $\alpha=O(1)$, highlighting a refined view of large-field behavior beyond the usual Swampland criteria. The results point to intriguing links between monodromic moduli spaces, hyperbolic geometry of congruence domains, and potential pathways to trans-Planckian axion physics in string theory, with prospects for extension to Calabi–Yau moduli spaces.

Abstract

We investigate how super-Planckian axions can arise when type IIB 3-form flux is used to restrict a two-axion field space to a one-dimensional winding trajectory. If one does not attempt to address notoriously complicated issues like Kahler moduli stabilization, SUSY-breaking and inflation, this can be done very explicitly. We show that the presence of flux generates flat monodromies in the moduli space which we therefore call 'Monodromic Moduli Space'. While we do indeed find long axionic trajectories, these are non-geodesic. Moreover, the length of geodesics remains highly constrained, in spite of the (finite) monodromy group introduced by the flux. We attempt to formulate this in terms of a 'Moduli Space Size Conjecture'. Interesting mathematical structures arise in that the relevant spaces turn out to be fundamental domains of congruence subgroups of the modular group. In addition, new perspectives on inflation in string theory emerge.

Figures (10)

• Figure 1: Winding flat direction of total length $\sim Nf$ (shown for $N=5$).
• Figure 2: A fundamental domain of the congruence subgroup $\Gamma^0(5)$ as a subset of the upper complex half plane is shown. The central strip without the 'triangle' touching the real axis corresponds to the standard fundamental domain of the complex structure modulus of a torus.
• Figure 3: The lower part of the fundamental domain of the congruence subgroup $\Gamma^0(7)$ is shown. Appropriate identifications of the boundaries are indicated Verrill:2001.
• Figure 4: A qualitative picture of a fundamental domain of a congruence subgroup as a Riemann surface. The throats correspond to the cusps of the fundamental domain together with the point at infinity.
• Figure 5: A fundamental domain of the congruence subgroup $\Gamma^0(12)$ with several cusps is shown.