Related papers: Relative quantum cohomology

We compute the open Gromov-Witten disk invariants and the relative quantum cohomology of the Chiang Lagrangian $L_\triangle \subset \mathbb{C}P^3$. Since $L_\triangle$ is not fixed by any anti-symplectic involution, the invariants may…

We define genus zero open Gromov-Witten invariants with boundary and interior constraints for a Lagrangian submanifold of arbitrary even dimension. The definition relies on constructing a canonical family of bounding cochains that satisfy…

This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the…

We construct open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians, which are weakly unobstructed by a bounding cochain. This uses the foundational work of…

We present a solution to the problem of defining genus zero open Gromov-Witten invariants with boundary constraints for a Lagrangian submanifold of arbitrary dimension. Previously, such invariants were known only in dimensions $2$ and $3$…

We prove a quantum version of Kalkman's wall-crossing formula comparing Gromov-Witten invariants on geometric invariant theory (git) quotients related by a change in polarization. The wall-crossing terms are gauged Gromov-Witten invariants…

We compute the quantum cohomology relative to a Lagrangian submanifold in some complete intersections. For quadric hypersurfaces, we also give a full computation of the genus zero open Gromov-Witten invariants.

This paper classifies separated bounding pairs for Lagrangian submanifolds that are homologically trivial inside the ambient space, under the assumption that restriction on cohomology from the ambient space to the Lagrangian is surjective.…

Given a closed symplectic manifold $X$, we construct Gromov-Witten-type invariants valued both in (complex) $K$-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is $K_p(n)$-local for some Morava $K$-theory $K_p(n)$.…

We define a new Gromov-Witten theory relative to simple normal crossing divisors as a limit of Gromov-Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism,…

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there…

We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane…

In algebraic geometry, Gromov--Witten invariants are enumerative invariants that count the number of complex curves in a smooth projective variety satisfying some incidence conditions. In 2001, A. Givental and Y.P. Lee defined new…

In the paper, we study the wall-crossing phenomenon of reduced open Gromov-Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open…

Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X, L; Z) with vanishing boundary in H_1 (L; Z), we prove that the algebraic number of J-holomorphic…

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of $X=\mathcal{O}_{\mathbb{P}^{1}}(-1,-1)$ to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero,…

We extend B. Hassett's theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear…

The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we…

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X, \omega)$. We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$…

We consider Open Gromov-Witten invariants for noncompact Calabi-Yau in the case the Lagrangian has the topology of $\R^2 \times S^1$. The definition of the invariant involves the choice of a frame for the Lagrangian, in accord with string…