The Swampland Distance Conjecture (SDC) restricts the geodesic distances that scalars can traverse in effective field theories as they approach points at infinite distance in moduli space. We propose that, when applied to the subset of light fields in effective theories with scalar potentials, the SDC restricts the amount of non-geodesicity allowed for trajectories along valleys of the potential. This is necessary to ensure consistency of the SDC as a valid swampland criterium at any energy scale across the RG flow. We provide a simple description of this effect in moduli space of hyperbolic space type, and products thereof, and obtain critical trajectories which lead to maximum non-geodesicity compatible with the SDC. We recover and generalize these results by expressing the SDC as a new Convex Hull constraint on trajectories, characterizing towers by their scalar charge to mass ratio in analogy to the Scalar Weak Gravity Conjecture. We show that recent results on the asymptotic scalar potential of flux compatifications near infinity in moduli space precisely realize these critical amounts of non-geodesicity. Our results suggest that string theory flux compactifications lead to the most generic potentials allowing for maximum non-geodesicity of the potential valleys while respecting the SDC along them.
Laplacians in Various Dimensions and the Swampland
Departamento de Física Teórica and Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid（马德里自治大学理论物理系和UAM/CSIC理论物理研究所）；Enrico Fermi Institute and Kadanoff Center for Theoretical Physics, University of Chicago（芝加哥大学恩里科·费米研究所和卡达诺夫理论物理中心）；Kavli Institute for Cosmological Physics, University of Chicago（芝加哥大学卡弗里宇宙物理学研究所）
