  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 SL(2,Z). 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 give 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.
