GLSM's for gerbes (and other toric stacks)

TL;DR

This paper develops gauged linear sigma models for toric stacks, demonstrating their physical equivalence, exploring mirror symmetry, and proposing a generalized Batyrev mirror conjecture, thus linking physics and advanced algebraic geometry.

Contribution

It introduces GLSMs for toric stacks, confirms their IR physics independence from presentations, and extends mirror symmetry techniques to stacks, including a new class of CFTs.

Findings

Gauged linear sigma models accurately describe toric stacks.

IR physics of different presentations of the same stack are equivalent.

New CFTs involving fields valued in roots of unity were discovered.

Abstract

In this paper we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) ${\bf C}^{\times}$ quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks, and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also check in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFT's to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using Morrison-Plesser-Hori-Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev's mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFT's, involving fields valued in roots of unity.