We compute open GW invariants for $\mathcal{K}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$, open orbifold GW invariants for $[\C^3/\Z_2]$, formulate an open crepant resolution conjecture and verify it for this pair. We show that open invariants can be glued together to deduce the Bryan-Graber closed crepant resolution conjecture for the orbifold $[\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)/\Z_2]$ and its crepant resolution $\mathcal{K}_{\mathbb{P}^1\times\mathbb{P}^1}$.
We study the orbifold Gromov-Witten theory of the quotient C^3/Z_3 in all genera. Our first result is a proof of the holomorphic anomaly equations in the precise form predicted by B-model physics. Our second result is an exact crepant resolution correspondence relating the Gromov-Witten theories of C^3/Z_3 and local CP2. The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.
We compute the relative orbifold Gromov-Witten invariants of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$, with respect to vertical fibers. Via a vanishing property of the Hurwitz-Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton's formula for the double ramification cycle, and the orbifold quantum Riemann-Roch. As a result parallel to its crepant resolution counterpart for $\mathcal{A}_n$, the GW/DT/Hilb/Sym correspondence is established for $[\mathbb{C}^2/\mathbb{Z}_{n+1}]$. The computation also implies the crepant resolution conjecture for relative orbif…
We derive a recursive formula for certain relative Gromov-Witten invariants with maximal tangency condition via the Witten-Dijkgraaf-Verlinde-Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.
Theta functions were defined for varieties with effective anticanonical divisor and are related to certain punctured Gromov-Witten invariants. In this paper we show that in the case of a log Calabi-Yau surface (X,D) with smooth very ample anticanonical divisor we can circumvent the notion of punctured Gromov-Witten invariants and relate theta functions and their multiplicative structure to certain 2-marked log Gromov-Witten invariants. This is a natural extension of the correspondence between wall functions and 1-marked log Gromov-Witten invariants.
We use Pixton's relations to prove a reconstruction theorem for genus 2 Gromov-Witten invariants in the style of Kontsevich-Manin (genus 0) and Getzler (genus 1). We also calculate genus 2 (descendant) Gromov-Witten invariants of $\mathbb{P}^2$ blown up at a finite number of points in general position.
We extend the definition of relative Gromov--Witten invariants with negative contact orders to all genera. Then we show that relative Gromov--Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.
Seiberg duality conjecture asserts that the Gromov-Witten theories (Gauged Linear Sigma Models) of two quiver varieties related by quiver mutations are equal via variable change. In this work, we prove this conjecture for $A_n$ type quiver varieties.
Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.
Connected Gromov-Witten invariants of [Sym^n(A_r)]
本文研究了对称积栈[Sym^n（A_r）]的连通Gromov-Witten不变量理论。我们导出了所有具有两个插入的等变不变量的封闭形式表达式，并揭示了该理论与三重A_r x P^1的相对Gromov-Witten理论之间的自然对应。当n小于或等于3时，我们确定了[Sym^n（A_1）]的三点（通常）Gromov-Witten不变量.
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 invariants of $X_{D,r}$. We also provide a new proof for the equality between genus zero relative and orbifold G…
We define the Gromov-Witten invariants for the parabolic bundles over an orbifold $C$ in various situation. Those bring us to refine this notion to get an accurate computation of the number of maximal subbundles of a sufficiently general parabolic bundle by means of the Intriligator-Varfa formula.
We study the cycle-valued reduced Gromov-Witten theory of a nonsingular projective K3 surface. For primitive curve classes, we prove that the correspondence induced by the reduced virtual fundamental class respects the tautological rings. Our proof uses monodromy over the moduli space of K3 surfaces, degeneration formulae and virtual localization. As a consequence of the monodromy argument, we verify an invariance property for Gromov-Witten invariants of K3 surfaces in primitive curve class conjectured by Oberdieck-Pandharipande.
