On \(A\)-twisted moduli stack for curves from Witten's gauged linear sigma models (Q707474)

The article presents an algebraic, functorial description of \textit{E.~Witten}'s gauged linear sigma models. These have been introduced in [Nucl. Phys. B 403, 159--222 (1993; Zbl 0910.14020)], and further investigated by \textit{D.~Morrison} and \textit{M.~Plesser} in [Nucl. Phys. B 440, 279--354 (1995; Zbl 0908.14014)] in order to compute generating functions for sigma models with target toric varieties, and Calabi-Yau hypersurfaces in these. The authors adopt \textit{D.~Cox}'s point of view in [Tôhoku Math. J., II. Ser. 47, 251--262 (1995; Zbl 0828.14035)], where he proves that the smooth toric variety \(X_\Delta\) corresponding to a fan \(\Delta\) is a fine moduli space for the functor of \(\Delta\)-collections. Using this idea, they define the \(A\)-twisted moduli stack \(\mathcal{AM}_g(X_\Delta)\) of genus \(g\) corresponding to \(X_\Delta\) by associating to a family \(\mathcal C\rightarrow U\) of quasi-stable curves of genus \(g\) the set of weak \(\Delta\)-collections on \(\mathcal C\). The first result of the paper states that \(\mathcal{AM}_g(X_\Delta)\) is an Artin stack. The authors proceed with a more detailed study of the genus zero case. They prove that \(\mathcal{AM}_0(X_\Delta)\) is isomorphic to a quotient stack, and this in turn is coarsely represented by a union of toric varieties, indexed by the elements of \(H_2(X_\Delta,\mathbb Z)\). Finally they prove, for smooth and convex \(X_\Delta\), the existence of a natural collapsing morphism of stacks \(\overline{\mathcal M}_{0,0}(\mathbb P^1 \times X_\Delta,(1,d)) \rightarrow \mathcal{AM}_0(X_\Delta)_d\), where \(d\in H_2(X_\Delta,\mathbb Z)\). The article ends with a survey of Witten's gauged linear sigma models.