#### Volume 22, issue 1 (2018)

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A mathematical theory of the gauged linear sigma model

### Huijun Fan, Tyler J Jarvis and Yongbin Ruan

Geometry & Topology 22 (2018) 235–303
##### Abstract

We construct a mathematical theory of Witten’s Gauged Linear Sigma Model (GLSM). Our theory applies to a wide range of examples, including many cases with nonabelian gauge group.

Both the Gromov–Witten theory of a Calabi–Yau complete intersection $X$ and the Landau–Ginzburg dual (FJRW theory) of $X$ can be expressed as gauged linear sigma models. Furthermore, the Landau–Ginzburg/Calabi–Yau correspondence can be interpreted as a variation of the moment map or a deformation of GIT in the GLSM. This paper focuses primarily on the algebraic theory, while a companion article will treat the analytic theory.

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##### Keywords
gauged linear sigma model, mirror symmetry, Gromov–Witten, Calabi–Yau, Landau–Ginzburg
##### Mathematical Subject Classification 2010
Primary: 14D23, 14L24, 14N35, 53D45, 81T60
Secondary: 14J32, 14L30, 32G81, 81T40