Related papers: Open WDVV equations and Virasoro constraints
We give a computability result for open Gromov-Witten invariants based on open WDVV equations. This is analogous to the result of Kontsevich-Manin for closed Gromov-Witten invariants. For greater generality, we base the argument on a formal…
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 construction of an open analogue of total descendant and total ancestor potentials via an "open version" of Givental's action. Our construction gives a genus expansion for an arbitrary solution to the open WDVV equations…
The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution…
Virasoro constraints for orbifold Gromov-Witten theory are described. These constraints are applied to the degree zreo, genus zero orbifold Gromov-Witten potentials of the weighted projective stacks $\mathbb{P}(1,N)$, $\mathbb{P}(1,1,N)$…
The WDVV equation is satisfied by the genus 0 correlation functions of any topological field theory in two dimensions coupled to topological gravity, and may be used to determine the genus 0 (rational) Gromov-Witten invariants of many…
The Virasoro conjecture is a conjectured sequence of relations among the descendent Gromov-Witten invariants of a smooth projective variety in all genera; the only varieties for which it is known to hold are a point (Kontsevich) and…
In this paper, we construct a pair of solutions to the open WDVV equations associated with the infinite-dimensional Frobenius manifolds that underlie the genus-zero universal Whitham hierarchy, and for the resulting flat F-manifolds, we…
The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the…
For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\leq 1$ Virasoro conjecture of…
Let $X$ be a smooth projective surface and $D$ a smooth rational ample divisor in $X$. We prove an all-genus generalization of the genus $0$ WDVV equation for primary Gromov--Witten invariants of the local 3-fold $\mathcal{O}_X(-D)$. The…
We state and prove a topological recursion relation that expresses any genus-g Gromov-Witten invariant of a projective manifold with at least a (3g-1)-st power of a cotangent line class in terms of invariants with fewer cotangent line…
Virasoro constraints are applied to degree zero Gromov-Witten theory of weighted projective stacks $\mathbb{P}(1,N)$ and $\mathbb{P}(1,1,N)$ to obtain formulas of descendant cyclic Hurwitz-Hodge integrals in higher genera.
Consider the genus one Hurwitz space $\mathcal{H}_1(n_0,\dots,n_m)$ of ramified covering of fixed degree with $m+1$ prescribed poles of order $n_0+1,\dots,n_m+1$, respectively. Based on a recent formula proved in \cite{Rejeb23}, we derive…
The Virasoro conjecture proposed by Eguchi-Hori-Xiong and S. Katz predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra.…
In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projective varieties. This result provides new evidence for the…
A new codimension 2 relation among descendent strata in the moduli space of stable, 3-pointed, genus 2 curves is found. The space of pointed admissible double covers is used in the calculation. The resulting differential equations satisfied…
We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the $\tau$-function of an open KdV heirarchy. A relation between the integrals and a…
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…
These brief notes record our puzzles and findings surrounding Givental's recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a projective line,…