Quantum cohomology of toric orbifolds

Abstract

For any compact toric orbifold (smooth proper Deligne-Mumford toric stack) $ Y$ with projective coarse moduli space, we show that the quantum cohomology $ QH(Y)$ is canonically isomorphic (in a formal neighborhood of a canonical bulk deformation) to a formal polynomial ring modulo the quantum Stanley-Reisner ideal introduced by Batyrev \cite{bat:qcr}. This generalizes results of Givental \cite{gi:eq}, Iritani \cite{iritani:conv} and Fukaya-Oh-Ohta-Ono \cite{fooo:tms} for toric manifolds and Coates-Lee-Corti-Tseng \cite{coates:wps} for weighted projective spaces. In the language of Landau-Ginzburg potentials, we identify $ QH(Y)$ with the ring of functions on the subset $ \Crit_+(W)$ of the critical locus $ \Crit(W)$ of an explicit potential $ W$ consisting of critical points mapping to the interior of the moment polytope, as in \cite{giv:hom}, \cite{iritani:conv}, \cite{fooo:tms} in the manifold case. Our proof uses algebro-geometric virtual fundamental classes, a quantum version of Kirwan surjectivity, and an equality of dimensions deduced using a toric minimal model program (tmmp). The previous cases treated by Givental \cite{gi:eq}, Iritani \cite{iritani:conv}, and Fukaya et al \cite{fooo:tms} used localization arguments for the residual torus action or open-closed Gromov Witten invariants. The existence of a Batyrev presentation implies that the quantum cohomology of $ Y$ is generically semisimple. This is related by a conjecture of Dubrovin, see \cite{bay:ss}, to the existence of a full exceptional collection in the derived category of $ Y$ proved by Kawamata \cite{kaw:der}, also using tmmps. Finally we discuss a connection with Hamiltonian non-displaceability. Any tmmp for $ Y$ with generic symplectic class defines a splitting of the quantum cohomology $ QH(Y)$ with summands indexed by flips, contractions or fibration in the tmmp, and each summand corresponds a collection of Hamiltonian non-displaceable Lagrangian tori in $ Y$ . In particular the existence of infinitely many tmmps can produce open families of Hamiltonian non-displaceable Lagrangians, such as in the examples in Wilson-Woodward \cite{ww:qu}.