Sums over topological sectors and quantization of Fayet-Iliopoulos parameters

TL;DR

The paper analyzes whether Fayet-Iliopoulos parameter quantization persists when moduli spaces are stacks, especially gerbes, connecting two-dimensional theories with altered topological sectors to four-dimensional field theories and string compactifications. It argues that 2D gerbe theories correspond to sigma models with restricted instantons, while 4D physics exhibits presentation dependence and may not flow to a nonlinear sigma model, necessitating stack-based universal descriptions. The work demonstrates that gerbe structures can be physically meaningful under certain conditions (topologically nontrivial spacetime or massive noninvariant states) and shows that fractional Bagger-Witten line bundles can arise, though they require careful interpretation within the IR framework. It also surveys topological defects, dualities, and consistency conditions for classical supergravity in the stack setting, highlighting implications for string theory and connections to geometric Langlands-type dualities.

Abstract

In this paper we discuss quantization of the Fayet-Iliopoulos parameter in supergravity theories with altered nonperturbative sectors, which were recently used to argue a fractional quantization condition. Nonlinear sigma models with altered nonperturbative sectors are the same as nonlinear sigma models on special stacks known as gerbes. After reviewing the existing results on such theories in two dimensions, we discuss examples of gerby moduli `spaces' appearing in four-dimensional field theory and string compactifications, and the effect of various dualities. We discuss global topological defects arising when a field or string theory moduli space has a gerbe structure. We also outline how to generalize results of Bagger-Witten and more recent authors on quantization issues in supergravities from smooth manifolds to smooth moduli stacks, focusing particular attention on stacks that have gerbe structures.