The Bagger-Witten line bundle is a line bundle over moduli spaces of two-dimensional SCFTs, related to the Hodge line bundle of holomorphic top-forms on Calabi-Yau manifolds. It has recently been a subject of a number of conjectures, but concrete examples have proven elusive. In this paper we collect several results on this structure, including a proposal for an intrinsic geometric definition over moduli spaces of Calabi-Yau manifolds and some additional concrete examples. We also conjecture a new criterion for UV completion of four-dimensional supergravity theories in terms of properties of the Bagger-Witten line bundle.
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