In this paper, we discuss Bagger–Witten line bundles over moduli spaces of SCFTs. We review how in general they are “fractional” line bundles, not honest line bundles, twisted on triple overlaps. We discuss the special case of moduli spaces of elliptic curves in detail. There, the Bagger–Witten line bundle does not exist as an ordinary line bundle, but rather is necessarily fractional. As a fractional line bundle, it is nontrivial (though torsion) over the uncompactified moduli stack, and its restriction to the interior, excising corners with enhanced stabilizers, is also fractional. It becomes an honest line bundle on a moduli stack defined by a quotient of the upper half plane by a metaplectic group, rather than [Formula: see text]. We review and compare to results of recent work arguing that well-definedness of the worldsheet metric implies that the Bagger–Witten line bundle admits a flat connection (which includes torsion bundles as special cases), and gives general arguments on the existence of universal structures on moduli spaces of SCFTs, in which superconformal deformation parameters are promoted to nondynamical fields ranging over the SCFT moduli space.
Vector bundles on the moduli stack of elliptic curves
