Related papers: A Mirror Theorem for Toric Stacks

We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A-infinity-algebras. The tangent…

For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow…

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the…

In this paper, we prove that there exists an equivalence between 2-category of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the 1-category of stacky fans. For this purpose, we obtain two main results. The first is to…

This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of…

We study the open/closed correspondence for the projective line via mirror symmetry. More explicitly, we establish a correspondence between the generating function of disk Gromov-Witten invariants of the complex projective line…

We study modularity properties of generating series of logarithmic Gromov-Witten invariants of elliptic fibrations relative to singular fibers. Motivated by predictions from Vafa-Witten theory, we conjecture that such generating series are…

This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $\mu_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual…

We propose a conjectural determination of the Gromov-Witten theory of a root stack along a smooth divisor. We verify our conjecture under an additional assumption.

The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of…

The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan's Gromov-Witten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here…

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line…

We study the relationship between derived categories of factorizations on gauged Landau-Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived…

Given a smooth target curve $X$, we explore the relationship between Gromov-Witten invariants of $X$ relative to a smooth divisor and orbifold Gromov-Witten invariants of the $r$-th root stack along the divisor. We proved that relative…

We establish a Hirzebruch-Riemann-Roch type theorem and Grothendieck-Riemann-Roch type theorem for matrix factorizations on quotient Deligne-Mumford stacks. For this we first construct a Hochschild-Kostant-Rosenberg type isomorphism…

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants…

Let X be a smooth elliptic fibration over a smooth base B. Under mild assumptions, we establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an O^* gerbe over a genus one fibration which is a…

We prove that for a compact toric manifold whose anti-canonical divisor is numerically effective, the Lagrangian Floer superpotential defined by Fukaya-Oh-Ohto-Ono is equal to the superpotential written down by using the toric mirror map…

Given a smooth toric variety X and an ample line bundle O(1), we construct a sequence of Lagrangian submanifolds of (C^*)^n with boundary on a level set of the Landau-Ginzburg mirror of X. The corresponding Floer homology groups form a…

Let $X$, $Y$ be smooth projective varieties over $\mathbf{C}$. Let $K$ be a bounded complex of coherent sheaves on $X\times Y$ and let $\Phi_K \colon \mathsf{D}^b_{\mathsf{Coh}}(X) \to \mathsf{D}^b_{\mathsf{Coh}}(Y)$ be the resulting…