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# Advances in Theoretical and Mathematical Physics

## Volume 10 (2006)

### Number 1

### G{LSM}s for gerbes (and other toric stacks)

Pages: 77 – 121

DOI: http://dx.doi.org/10.4310/ATMP.2006.v10.n1.a4

#### Authors

#### 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 see 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 CFTs 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 CFTs, involving fields valued in roots of unity.

#### 2010 Mathematics Subject Classification

Primary 81T30. Secondary 14M25, 14N35.