A Mathematical Theory of the Gauged Linear Sigma Model

TL;DR

This paper develops a rigorous mathematical framework for Witten's Gauged Linear Sigma Model, unifying various theories like Gromov-Witten and FJRW, and exploring their interrelations through geometric invariant theory deformations.

Contribution

It introduces a comprehensive algebraic construction of GLSM applicable to non-Abelian groups, linking Landau-Ginzburg and Calabi-Yau theories within a unified model.

Findings

Unified algebraic framework for GLSM

Representation of Gromov-Witten and FJRW theories as GLSM

Interpretation of Landau-Ginzburg/Calabi-Yau correspondence as GIT deformation

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 non-Abelian 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.