Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack X_{D,r}. As an application, we also obtain an I-function for relative Gromov--Witten theory following the relation between relative and orbifold Gromov--Witten invariants.

We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

We study here the crepant resolution correspondence for the torus equivariant descendent Gromov-Witten theories of Hilb(C2) and Sym(C2).The descendent correspondence is obtained from our previous matching of the associated CohFTs by applying Givental's quantization formula to a specific symplectic transformation K. The first result of the paper is an explicit computation of K. Our main result then establishes a fundamental relationship between the Fourier-Mukai equivalence of the associated derived categories (by Bridgeland, King, and Reid) and the symplectic transformation K via Iritani's integral structure. The results use Haiman's Fourier-Mukai calculations and are exactly aligned with Iritani's point of view on crepant resolution.

Given a smooth projective variety X and a smooth divisor D\subset X. We study relative Gromov-Witten invariants of (X,D) and the corresponding orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, we prove that orbifold Gromov- Witten invariants of X_{D,r} are polynomials in r. Moreover, higher genus relative Gromov-Witten invariants of (X,D) are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invari- ants of X_{D,r}. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise. When r is sufficiently large and X=C is a curve, we prove that stationary relative invariants of C are equal to the stationary orbifold invariants in all genera.

Roughly speaking, to any space M with perfect obstruc- tion theory we associate a space N with symmetric perfect obstruction theory. It is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space M we associate its (−1)-shifted cotangent bundle N. By localising from N to its C∗-fixed locus M this gives five notions of virtual signed Euler characteristic of M: (1) TheCiocan-Fontanine-Kapranov/Fantechi-G ̈ottschesignedvirtual Euler characteristic of M defined using its own obstruction theory, (2) Graber-Pandharipande’s virtual Atiyah-Bott localisation of the virtual cycle of N to M, (3) Behrend’s Kai-weighted Euler characteristic localisation of the vir- tual cycle of N to M, (4) Kiem-Li’s cosection localisation of the virtual cycle of N to M, (5) (−1)vd times by the topological Euler characteristic of M. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.