The moduli space in the gauged linear sigma model

Abstract

Abstract. This is a survey article for the mathematical theory of Witten's Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper we focus on the description of the moduli.

Stable gauged maps

“…For example, if X = P 4 with weights 0, 1, 1, −1, −1 then X/ / t G undergoes an Atiyah flop as t passes through zero. Using C × -equivariant versions of the gauged Gromov-Witten invariants we obtain an explicit formula for the difference (22) τ X/ /+G κ G X,+ − τ X/ /−G κ G X,− in either cohomology or K-theory. We say that the variation is crepant if the weights at the G-fixed points at t = 0 sum to zero; in this case the birational equivalence between X/ / − G and X/ / + G is a combination of flops.…”

“…The splitting axiom for the gauged invariants is somewhat different than the usual splitting axiom in Gromov-Witten theory: the potential τ G X is a non-commutative version of a trace on the Frobenius manifold QH G (X). Note that there are several other notions of gauged Gromov-Witten invariants, for example, Ciocan-Fontanine-Kim-Maulik [15], Frenkel-Teleman-Tolland [23], as well as a growing body of work on gauged Gromov-Witten theory with potential [66], [22].…”

Geometry of gauged LG-model with compatible boundary conditions

“…Under GLSM, the LG/CY correspondence can be interpreted as a variation of the moment map µ. Lately, Fan-Jarvis-Ruan constructed a rigorous mathematical theory for GLSM [9,10].…”