A. Zinger
Following a question of K. Hori at K. Fukaya's 60th birthday conference, we relate the recently established WDVV-type relations for real Gromov-Witten invariants to topological recursion relations in a real setting. We also describe precisely the connections between the relations themselves previously observed by A. Alcolado.
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WDVV-based recursion for open Gromov-Witten invariants
本文基于WDVV方程研究开放Gromov-Witten不变量的可计算性，并证明了射影空间乘积上的不变量消失性质。
WDVV-Type Relations for Welschinger's Invariants: Applications
本文回顾了实辛流形中Welschinger不变量的Solomon关系与WDVV型关系，并证明在射影空间等重要情形下这些关系可提供完整递归以确定所有不变量，进而利用Mathematica计算出低次不变量表格，为实有理曲线计数提供了下界。
In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov-Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This note provides another, more complex-geometric, proof of the latter. The main part of this approach readily extends to real symplectic manifolds with empty real locus, but not to the general case.
We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all c…
Steenrod Pseudocycles, Lifted Cobordisms, and Solomon's Relations for Welschinger's Invariants
本文通过从稳定实曲线的Deligne-Mumford模空间提升同调关系，为实辛四维流形的盘不变量建立了两条WDVV型关系，进而导出Solomon于2007年宣布的Welschinger不变量递推关系，并提供了一个通过不可定向空间间态射提升关系的通用拓扑框架。
Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants
本文基于重构猜想与对数拓扑递归等新进展，推导了条形几何环面Calabi-Yau三维流形的开拓扑弦配分函数及量子曲线，阐明其与开DT配分函数等的关联，并通过实例说明3D-5D墙穿越的影响。
We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. …
We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is $x = \frac{1}{2}y^2$, the same as spectral curve used to calculate intersection numbers for closed Riemann surfaces, but the formula itself is a variation of the usual Eynard-Orantin recursion. It looks like the recursion formula used for spectral curves of degree 3, and also includes features present in $β$-deformed models. The recursion formula suggests a conjectural refinement to the generating function that allows for distinguishing…
We construct locally defined symplectic torus actions on ribbon graph complexes. Symplectic reduction techniques allow for a recursive formula for the symplectic volumes of these spaces. Taking the Laplace transform results in the Eynard-Orantin recursion formulas for the Airy curve x = y^2 / 2.
We give an Eynard-Orantin type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon graphs. As an application we obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and the symplectic volumes of the moduli space of curves.
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 analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.
It was pointed out by Eliashberg in his ICM 2006 plenary talk that the integrable systems of rational Gromov-Witten theory very naturally appear in the rich algebraic formalism of symplectic field theory (SFT). Carefully generalizing the definition of gravitational descendants from Gromov-Witten theory to SFT, one can assign to every contact manifold a Hamiltonian system with symmetries on SFT homology and the question of its integrability arises. While we have shown how the well-known string, dilaton and divisor equations translate from Gromov-Witten theory to SFT, the next step is to show ho…
The ABCD of topological recursion
本文研究了规范群作用下的量子Airy结构的分类问题，提供李代数工具并给出${\rm dim}\\,V=2$时的分类，同时描述了来自Frobenius代数等四类结构的量子Airy结构。
Based on the localization result for descendants in rational SFT moduli spaces from our last joint paper, we prove topological recursion relations for the Hamiltonian in SFT of symplectic mapping tori and in local SFT. Combined with the dilaton equation in SFT, we use them to prove a reconstruction theorem for descendants from primaries. While it turns out that (in contrast to Gromov-Witten theory and non-equivariant cylindrical contact homology) the descendant Hamiltonian cannot be computed from the Hamiltonian without descendants alone, we get that the only additional piece of information ne…
Topological recursion and variations of spectral curves for twisted Higgs bundles
本文针对扭 Higgs 丛提出保留亚纯奇性的扭拓扑递归，并证明零亏格 Eynard-Orantin 微分给出谱曲线周期矩阵的 Taylor 展开。
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