Bagger-Witten line bundles on moduli spaces of elliptic curves

TL;DR

This paper explores the fractional nature of Bagger-Witten line bundles on moduli spaces of elliptic curves, highlighting their nontriviality, and discusses conditions under which they become honest line bundles or admit flat connections.

Contribution

It provides a detailed analysis of Bagger-Witten line bundles on elliptic curve moduli spaces, emphasizing their fractional properties and conditions for being honest line bundles or flat.

Findings

Bagger-Witten line bundles are fractional and nontrivial over moduli stacks.

On a quotient of the upper half plane, they become honest line bundles.

Existence of flat connections relates to worldsheet metric well-definedness.

Abstract

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.