#### 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.

##### 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
##### Publication
Received: 2 November 2015
Revised: 23 December 2016
Accepted: 30 January 2017
Published: 31 October 2017
Proposed: Jim Bryan
Seconded: Richard Thomas, Simon Donaldson
##### Authors
 Huijun Fan School of Mathematical Science Beijing (Peking) University Beijing China Tyler J Jarvis Department of Mathematics Brigham Young University Provo, UT United States Yongbin Ruan Mathematics Department University of Michigan Ann Arbor, MI United States Beijing International Center for Mathematical Science Peking University Beijing China