Abstract
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 criterion 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.
Highlights
An important point in the discussion of Swampland Distance Conjecture (SDC) is that it should apply to adiabatic motion in moduli space
Given an effective theory violating the SDC in its moduli space, one could always argue that this corresponds to the theory on M, and that above certain scale Λ the theory is completed to a larger moduli space M which obeys it, and which can in principle be completed into a quantum gravity theory
In this paper we have discussed the interpretation of the Swampland Distance Conjecture in effective theories with scalar potentials leading to valleys of light fields
Summary
We focus our analysis on trajectories approaching points at infinity in moduli space, in the spirit of the SDC, since the interesting physics occurs in the asymptotic region near infinity. We discuss moduli spaces given by a hyperbolic plane, or products thereof Despite their apparent simplicity, they are key to describing moduli spaces of general CY compactifications near their boundaries at infinity [14, 17, 51], to the extent of encoding much of the dynamics of these models [59]. They are key to describing moduli spaces of general CY compactifications near their boundaries at infinity [14, 17, 51], to the extent of encoding much of the dynamics of these models [59] They allow for explicit computations which will be useful to motivate our generalizations in later sections
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