A few Ricci-flat stacks as phases of exotic GLSM's

TL;DR

The paper analyzes exotic GLSMs with a $U(1) \times \mathbb{Z}_2$ gauge group and a superpotential quadratic in $p$ fields, showing that the Calabi–Yau branched double cover phases described by HKM rely on the $\mathbb{Z}_2$ gerbe. It then proposes that removing the $\mathbb{Z}_2$ factor yields IR phases described by Ricci-flat stacks that resemble Fano manifolds with a codimension-one $\mathbb{Z}_2$ orbifold along the would-be branch locus, resulting in non-Calabi–Yau but Ricci-flat geometries. These stacks have 2-torsion canonical class and illuminate subtle covers-versus-quotients distinctions in GLSMs, including local versus global gerbe data and quantum symmetry considerations. The work also discusses implications for worldsheet supersymmetry, brane gradings, and the relation to Enriques-like geometry and noncommutative resolutions near singular loci, outlining avenues for further study of SCFTs and D-branes on Ricci-flat stacks.

Abstract

In this letter we follow up recent work of Halverson-Kumar-Morrison on some exotic examples of gauged linear sigma models (GLSM's). Specifically, they describe a set of U(1) x Z_2 GLSM's with superpotentials that are quadratic in p fields, rather than linear as is typically the case. These theories RG flow to sigma models on branched double covers, where the double cover is realized via a Z_2 gerbe. For that gerbe structure, and hence the double cover, the Z_2 factor in the gauge group is essential. In this letter we propose an analogous geometric understanding of phases without that Z_2, in terms of Ricci-flat (but not Calabi-Yau) stacks which look like Fano manifolds with hypersurfaces of Z_2 orbifolds.