From Frame Covariance to the Swampland Distance Conjecture

TL;DR

The paper addresses the frame-dependence of field-space geometry in gravitational EFTs and its impact on Swampland Distance Conjectures. By promoting the conformal factor to a dynamical field, it constructs a frame-augmented field space \mathcal{M}^\omega in which all conformal frames are foliations, with Einstein frame singled out as a totally geodesic, geodesic-preserving foliation. This framework yields a Weyl- and unit-covariant definition of the field-space metric \mathcal{G}_{ij} and enables a covariant ADM-like treatment of geodesics and distances across frames. Applying this to toroidal compactifications, the authors derive frame-covariant versions of the Species Scale Distance Conjecture and the Sharpened Distance Conjecture, showing that their bounds follow from universal properties of gravitational EFTs under Weyl transformations rather than from quantum gravity specifics. Overall, the work clarifies the role of frame covariance in the Distance Conjectures and provides a general toolkit to extend these ideas to a broader class of scalar-tensor theories.

Abstract

Field space geometry plays a central role within the Swampland Programme, most notably in the various Distance Conjectures. However, for gravitational EFTs, this geometry is not uniquely defined: one can cast the action in many synonymous descriptions related by Weyl transformations, in which the field space metric transforms non-trivially across conformal frames. This raises a crucial question of how we are meant to think of the field space metric in view of employing the Swampland Conjectures. In this work we resolve this ambiguity by developing a fully frame-covariant framework for studying gravitational EFTs. We show that all conformal frames arise as distinct foliations of a singular higher-dimensional auxiliary geometry. Applying ADM formalism to the augmented field space, it is clear how Weyl- and unit transformations can be understood from a geometric point of view. Using this framework, we revisit the Species Scale Distance Conjecture and Sharpened Distance Conjecture, and show how the bounds derive from universal properties of gravitational EFTs under Weyl transformations. This strongly suggests that aspects of these conjectures apply to a much broader class of scalar-tensor theories and are consequences of frame covariance, rather than constraints imposed by quantum gravity.

Figures (3)

• Figure 1: Illustration of the augmented field space $\mathcal{M}^\omega$ and different conformal frames $\mathsf{\Sigma},\mathsf{\Sigma}'$ as codimension one hypersurfaces. Every hypersurface gives rise to a foliation of frames related by constant Weyl transformations. A frame transformation $\mathcal{F}$ corresponds to a map sending leaves to leaves between different foliations.
• Figure 2: If $\gamma$ is a spacelike geodesic with respect to the augmented field space metric $\mathsf{G}_{IJ}$, then it will also be a geodesic on Einstein frame. Under a frame transformation $\gamma\mapsto \widetilde{\gamma}$ it is no longer guaranteed to be a geodesic.
• Figure 3: Sketching the $d$-dimensional moduli basis $(\omega,\boldsymbol{\phi})$ and $D$-dimensional moduli basis $(\varpi,\boldsymbol{\rho})$ in the frame-augmented field space. For an inherently higher-dimensional theory originally in Einstein frame before compactification, the timelike direction is $\varpi$ rather than $\omega$.