Quantum Kirwan morphism and Gromov-Witten invariants of quotients. II (Q498948)

\textit{F. C. Kirwan} [Cohomology of quotients in symplectic and algebraic geometry. Princeton, New Jersey: Princeton University Press (1984; Zbl 0553.14020)] studied the map from the equivariant cohomology of a Hamiltonian group action to the cohomology of the symplectic quotient. The paper under review deals with the quantum version of this situation. Let \(X\) be a smooth projectively embedded variety with a connected reductive group action such that stable locus is equal to the semistable locus. Let \(QH(X/\!/G)\) be the quantum cohomology of the GIT quotient \(X/\!/G\), and let \(QH_G(X)= H_G(X)\otimes \Lambda_X^G\) denote the equivariant quantum cohomology where \( \Lambda_X^G \subset\mathrm{Hom}(H_2^G(X,\mathbb{Z}))\) is Novikov field. \textit{A. B. Givental} [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)] equipped the latter with a product structure arising from the equivariant Gromov-Witten theory. \textit{E. Gonzalez} and and the author of the paper under review [Math. Z. 273, No. 1--2, 485--514 (2013; Zbl 1258.53092)] proved that, under some conditions, the moduli space of stable gauged maps from a smooth connected projective curve is a proper Deligne-Mumford stack equipped with a perfect obstruction theory which leads to the definition of the Gauged Gromov-Witten invariants. The paper under review constructs virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-Witten potential. This is the second paper in a sequence of papers aiming at constructing a quantum version of the Kirwan map. The first result of the series of papers [\textit{C. T. Woodward}, Transform. Groups 20, No. 2, 507--556 (2015; Zbl 1326.14134)] is the construction of ``Quantum Kirwan Morphism'' \[ \kappa_X^G:QH_G(X)\to QH(X/\!/G) \] by means of the virtual integration over a compactified stack of affine gauged maps. \(\kappa_X^G\) is a morphism of CohFT algebras, which can be considered as a non-linear generalization of an algebra homomorphism. The second main result of the series of the papers relates in the large area limit the graph potentials via the quantum Kirwan morphism. More precisely, suppose that \(C\) is a smooth projective curve such that all the semistable gauged maps from \(C\) to \(X\) are stable for sufficiently large stability parameters. Then \[ \tau_{X/\!/G}\circ\kappa^G_X=\lim_{\rho\to \infty}\tau_X^G \] where \(\rho\) is the stability parameter, \(\tau_{X/\!/G}\) genus zero graph potential for the GIT quotient \(X/\!/G\), and \(\tau^G_X\) is the gauged Gromov-Witten potential. Moreover, the localized version of this theorem that arises as the fixed point contributions for a circle acting on the domain. This localization in the case of Gromov-Witten invariants gives rise to a solution for a version of the Picard-Fuchs quantum differential equation for \(X/\!/G\). Some of these results have overlaps and connections with the works of Givental, Lian-Liu-Yau, Iritani, Ciocan-Fontanine-Kim-Maulik, and Coates-Corti-Iritani-Tseng who have taken different approaches.