Abstract:
We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya-Oh-Ohta-Ono and Cho-Oh on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without virtual fundamental chains. Finally we describe a conjectural relationship to Floer cohomology in the quotient.

Abstract:
We show that Tolman's example (of a six dimensional Hamiltonian $T^2$-space with isolated fixed points and no compatible K\"{a}hler structure) can be constructed from the flag variety $U(3)/U(1)^3$ by $U(2)$-equivariant symplectic surgery. This implies that Tolman's space has a ``transversal multiplicity-free'' action of $U(2)$ and that Delzant's theorem ``every compact multiplicity-free torus action is K\"{a}hler'' \cite{D1} does not generalize to non-abelian actions.

Abstract:
For smooth projective G-varieties, we equate the gauged Gromov-Witten invariants for sufficiently small area and genus zero with the invariant part of equivariant Gromov-Witten invariants. As an application we deduce a gauged version of abelianization for Gromov-Witten invariants. In the symplectic setting, we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map.

Abstract:
For any compact symplectic toric orbifold Y, we show that the quantum cohomology QH(Y) is isomorphic to a formal polynomial ring modulo the quantum Stanley-Reisner ideal introduced by Batyrev at a canonical bulk deformation. This generalizes results of Givental, Iritani and Fukaya-Oh-Ohta-Ono for toric manifolds and Coates-Lee-Corti-Tseng 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 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 existence of a Batyrev presentation implies that the quantum cohomology of Y is generically semisimple. This is related by a conjecture of Dubrovin, to the existence of a full exceptional collection in the derived category of Y proved by Kawamata, 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 transitions 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.

Abstract:
We prove a quantum version of the localization formula of Witten which relates invariants of a git quotient with the equivariant invariants of the action. As an application, we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus; this is a version of the "abelian/non-abelian" conjecture of Bertram, Ciocan-Fontanine, and Kim. As sample applications we give a formula for a solution to the quantum differential equation (qde) for the moduli space of points on the projective line and a solution to the twisted qde for Atiyah-Drinfeld-Hitchin-Manin quiver moduli.

Abstract:
We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical $C^1$-orbifold structure.

Abstract:
This is the third in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. We also give a formula for a solution to the quantum differential equation in terms of a localized gauged potential. These results overlap with those of Givental, Lian-Liu-Yau, Coates-Corti-Iritani-Tseng and Ciocan-Fontanine-Kim.

Abstract:
This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. In this part we construct virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-Witten potential.

Abstract:
We prove a multiplicity formula for Riemann-Roch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a Riemann surface.

Abstract:
In this version referee's comments have been incorporated. Besides minor corrections, new material has been added on irrational markings. To appear in Ann. Inst. Fourier. We prove a condition for the existence of flat bundles on the punctured two-sphere with prescribed holonomies around the punctures, involving Gromov-Witten invariants of generalized flag varieties. This generalizes the case of special unitary bundles described by S. Agnihotri and the second author and independently P. Belkale.