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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 other than two. We will propose its nontriviality as a new criterion for existence of UV completions of four-dimensional supergravity theories. Most of the talk will focus on outlining statements, rather than giving rigorous arguments. However, at the end, 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.

Email tudor@math.ucdavis.edu for Zoom info/password.