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We present a first-principles derivation of the K\"ahler metric foraxion-like moduli of conformally Calabi-Yau compactifications of IIB stringtheory with imaginary self-dual 3-form flux at the classical level. We findthat the warp factor and flux modify the moduli space metric and thereforeK\"ahler potential even in classical supergravity, with the modificationsscaling as (volume)$^{-2/3}$ in the large-volume limit. Our derivationemphasizes the role of constraints from 10D gauge symmetries and highlightsmetric formality as a geometric property that protects the moduli space ofhighly supersymmetric toroidal orientifolds. Our results have importantquantitative implications for nonperturbative moduli stabilization,phenomenology, and cosmology in flux compactifications.

The Dimensional Reduction and K\"ahler Metric of Forms In Flux and Warping