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In this paper we discuss Bagger-Witten line bundles over moduli spaces ofSCFTs. We review how in general they are `fractional' line bundles, not honestline bundles, twisted on triple overlaps. We discuss the special case of modulispaces of elliptic curves in detail. There, the Bagger-Witten line bundle doesnot exist as an ordinary line bundle, but rather is necessarily fractional. Asa fractional line bundle, it is nontrivial (though torsion) over theuncompactified moduli stack, and its restriction to the interior, excisingcorners with enhanced stabilizers, is also fractional. It becomes an honestline bundle on a moduli stack defined by a quotient of the upper half plane bya metaplectic group, rather than SL(2,Z). We review and compare to results ofrecent work arguing that well-definedness of the worldsheet metric implies thatthe Bagger-Witten line bundle admits a flat connection (which includes torsionbundles as special cases), and give general arguments on the existence ofuniversal structures on moduli spaces of SCFTs, in which superconformaldeformation parameters are promoted to nondynamical fields ranging over theSCFT moduli space.

Bagger-Witten line bundles on moduli spaces of elliptic curves