Abstract: This talk will concern advances in understanding explicitly the Bagger-Witten line bundle appearing in four-dimensional N=1 supergravity, which is closely related to the Hodge line bundle on a moduli space of Calabi-Yaus. This has recently been a subject of interest, but explicit examples have proven elusive in the past. In this talk we will outline some recent advances, including (1) a description of the Bagger-Witten line bundle on a moduli space of Calabi-Yau's as a line bundle of covariantly constant spinors (resulting in a square root of the Hodge line bundle of holomorphic top-forms), (2) results suggesting that it (and the Hodge line bundle) is always flat, but possibly never trivial, over moduli spaces of Calabi-Yaus of maximal holonomy and dimension greater than two. We will propose its nontriviality as a new criterion for existence of UV completions of four-dimensional supergravity theories. If time permits, we will explicitly construct an example, to concretely display these properties, and outline results obtained with Ron Donagi and Mark Macerato for other cases.