Ezra Getzler
We study the Toda conjecture of Eguchi and Yang for the Gromov-Witten invariants of CP^1,using the bihamiltonian method of the formal calculus of variations. We also study its relationship to the Virasoro conjecture for CP^1, recently proved by Givental (math.AG/0108100).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Bihamiltonian tests for integrable systems associated to rank-$1$ F-CohFTs
本文研究了与秩为1的F-CohFT相关的DR层次结构的双辛测试，推测其形成2参数族，并发现其中1参数子族与Camassa-Holm层次结构Miura等价，同时提出与Degasperis-Procesi层次结构的关联。
A generalization of the mirror conjecture is proven for the manifolds of complete flags in C^n.
We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.
We study a reduction of the Toda lattice in the limit of infintesimal lattice spacing. Using this reduction, we formulate a conjecture for the equivariant Gromov-Witten invariants of the sphere, which we prove in genus 0. This conjecture follows from recent work of Okounkov and Pandharipande (math.AG/0207233), combined with results from the sequel to this paper.
We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras U_q(g'). The conjecture is proved in the case of classical flag manifolds of the series A. The proof is based on a refinement of the famous Atiyah-Hirzebruch argument for rigidity of arithmetical genus applied to hyperquot-scheme compactifications of spaces of rational curves in the flag manifolds.
We prove the Givental conjecture that the (equivariant) quantum cohomology of flag manifolds G/B are governed by the conservation law of Toda lattices. In addition, we find the quantum differential module structure of the flag manifolds.
Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. By two different wall-crossing arguments we prove two different explicit formulae relating rank 0 Donaldson-Thomas invariants (counting torsion sheaves on $X$ supported on ample divisors) in terms of rank 1 Donaldson-Thomas invariants (counting ideal sheaves of curves) and Pandharipande-Thomas invariants. In particular, we prove a slight modification of Toda's formulation of OSV conjecture for $X$. When $X$ is of Picard rank one, we also give an explicit formula for rank tw…
A proof of all ranks S-duality conjecture for K3 surfaces
利用Toda多重覆盖公式计算K3曲面上任意秩的SU(r)/Z_r-Vafa-Witten不变量，并基于Tanaka-Thomas结果证明所有秩的S-对偶猜想。
Let $\CP^1_{k,m}$ be the orbifold structure on $\CP^1$ obtained via uniformizing the neighborhoods of 0 and $\infty$ respectively by $z\mapsto z^k$ and $w\mapsto w^m.$ The diagonal action of the torus on the projective line induces naturally an orbifold action on $\CP^1_{k,m}.$ In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of $\CP^1_{k,m}$ satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equi…
We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V_5 and V_22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V_5 and V_22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are…
Rank $r$ DT theory from rank $1$
该研究针对满足特定猜想的Calabi-Yau 3-折叠，将任意秩r的Gieseker半稳定层的广义DT不变量用秩1层的不变量表示，且通过MNOP猜想证明其由Gromov-Witten不变量决定。
We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators in our construction are given in terms of the $K$-theory of $\mathbb{P}^1_{a_1,a_2,a_3}$ via Iritani's $Γ$-class modification of the Chern character map. We also identify our HQEs with an appropriate Kac--Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of $\mathbb{P}^1$ .
In math.AG/0207233, Okounkov and Pandharipande gave an operator formalism for computing the equivariant Gromov-Witten theory of the projective line. This thesis extends their result to orbifold lines. In the effective case the theory is again governed by the 2-Toda hierarchy. In the ineffective case the decomposition conjecture of hep-th/0606034 is verified.
In analogy with the Gopakumar-Vafa (GV) conjecture on Calabi-Yau (CY) 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In a joint work with Maulik and Toda, the author conjectured their genus zero invariants are $\mathrm{DT_4}$ invariants of one dimensional stable sheaves. In this paper, we study this conjecture on the total space of canonical bundle of a Fano 3-fold $Y$, which reduces to a relation between twisted GW and $\mathrm{DT_3}$ invariants on $Y$. Examples are computed for both compact and n…
A proof of GW/DT4 conjecture on local projective plane
本文证明了局部射影平面（$\mathcal{O}_{\mathbb{P}^2}(-1,-2)$全空间）上一维稳定层的DT4不变量对所有度数曲线类的闭公式，通过典范选取模空间方向，并结合GV型不变量验证了作者、Maulik和Toda的猜想。
Stable pair invariants of local Calabi-Yau 4-folds
本文研究局部 Calabi-Yau 四维流形的稳定对不变量，将其化为曲面上点的 Hilbert 概型的相交数，从而为 Cao-Maulik-Toda 猜想提供新的低度数验证，并揭示其与 Carlsson-Okounkov 数的联系。
Generalized Bogomolov--Gieseker inequalities via Frobenius
该研究针对维数≥3的光滑射影簇上的倾斜半稳定对象，建立了广义博戈莫洛夫-吉泽克不等式，利用弗罗贝尼乌斯态射的作用分析证明了卡斯泰尔诺沃界猜想，并结合相关工作得出卡拉比-丘三维簇上半稳定层的唐纳森-托马斯不变量为辛不变量的结论。
We introduce the notion of the (instanton part of the) Seiberg-Witten prepotential for general Schrodinger operators with periodic potential. In the case when the operator in question is integrable we show how to compute the prepotential in terms of period integrals; this implies that in the integrable case our definition of the prepotential coincides with the one that has been extensively studied in both mathematical and physical literature. As an application we give a proof of Nekrasov's conjecture connecting certain "instanton counting" partition function for an arbitrary simple group G wit…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。