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 [FJR16] will treat the analytic theory.
We introduce W -spin structures on a Riemann surface † and give a precise definition to the corresponding W -spin equations for any quasi-homogeneous polynomial W . Then we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W -spin equations when W D W .x 1 ; : : : ; x t / is a nondegenerate, quasi-homogeneous polynomial with fractional degrees (or weights) q i < 1 2 for all i . In particular, the compactness theorem holds for the superpotentials E 6 ; E 7 ; E 8 or A n 1 ; D nC1 for n 3.
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