Yi Li
In this paper we derive some new Hodge integral identities by taking limits of the Marino-Vafa formula. These identities include the formula of lambda_{1}lambda_{g}-integral on M_{g,1}, the vanishing result of lambda_{g}ch_{2l}(E)-integral on M_{g,1} for l taking value between 1 and g-3. Finally, we give another simple proof of lambda_{g} conjecture and some examples of low genus integral.
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We prove a generalized Mariño-Vafa formula for Hodge integrals over $\Mbar_{g, γ-μ}(\cB G)$ with $G$ an arbitrary finite abelian group. Then we use this formula to study the local Gromov-Witten theory of an orbi-curve with cyclic stack points in a Calabi-Yau three-orbifold.
We present a proof of the full Mariño-Vafa Conjecture that identifies certain open string invariants of the resolved conifold with the Chern-Simons knot invariant of the unknot, i.e. the quantum dimensions.
We outline a proof of a remarkable formula for Hodge integrals conjectured by Marino and Vafa in hep-th/0108064 base on large N duality.
We prove a remarkable formula for Hodge integrals conjectured by Marino and Vafa based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.
In this note we derive some consequences from the Mariño-Vafa formula and the cut-and-join equation, these include unified simple proofs of the $λ_g$ conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mariño-Vafa formula.
We prove a formula for certain cubic $\Z_n$-Hodge integrals in terms of loop Schur functions. We use this identity to prove the Gromov-Witten/Donaldson-Thomas correspondence for local $\Z_n$-gerbes over $\proj^1$.
By the same method introduced in [9], we calculate the Laplace transform of the celebrated cut-and-join equation of Mariño-Vafa formula discovered by C. Liu, K. Liu and J. Zhou [17]. Then, we study the applications of the polynomial identity (1) obtained in theorem 1.1 of this paper. We show the proof Bouchard-Mariño conjecture for C^3 which was given by L. Chen [5] Firstly. Subsequently, we will present how to obtain series Hodge integral identities from this polynomial identity (1). In particular, the main result in [9] is one of special case in such series of Hodge integral identities. At l…
Based on the duality between open-string theory on noncompact Calabi-Yau threefolds and Chern-Simons theory on three manifolds, M Marino and C Vafa conjectured a formula of one-partition Hodge integrals in term of invariants of the unknot (hep-th/0108064). Many Hodge integral identities, including the lambda_g conjecture and the ELSV formula, can be obtained by taking limits of the Marino-Vafa formula. Motivated by the Marino-Vafa formula and formula of Gromov-Witten invariants of local toric Calabi-Yau threefolds predicted by physicists, J Zhou conjectured a formula of two-partition Hodge int…
We prove some combinatorial results related to a formula on Hodge integrals conjectured by Mariño and Vafa. These results play important roles in the proof and applications of this formula by the author jointly with Chiu-Chu Melissa Liu and Kefeng Liu. We also compare with some related results on Hurwitz numbers and obtain some closed expressions for the generating series of Hurwitz numbers and the related Hodge integrals.
We prove the Gopakumar-Marino-Vafa formula for special cubic Hodge integrals. The GMV formula arises from Chern-Simons/string duality applied to the unknot in the three sphere. The GMV formula is a q-analog of the ELSV formula for linear Hodge integrals. We find a system of bilinear localization equations relating linear and special cubic Hodge integrals. The GMV formula then follows easily from the ELSV formula. An operator form of the GMV formula is presented in the last section of the paper.
We propose a conjectural formula expressing the generating series of some Hodge integrals in terms of representation theory of Kac-Moody algebras. Such generating series appear in calculations of Gromov-Witten invariants by localization techniques. It generalizes a formula conjectured by Mariño and Vafa, recently proved in joint work with Chiu-Chu Melissa Liu and Kefeng Liu. Some examples are presented.
We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections are computed by the same method. Finally, we present an application to the non-vanishing of the Pontrjagin ring of the moduli space of bundles.
We prove a closed formula for leading Gopakumar- Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some similar features with Goettsche-Yau-Zaslow formula: Connection with Hilbert schemes, connection with quasimodular forms, and quadratic property after suitable transformation. In Part I of this paper we will present the case of projective plane, more general cases will be presented in Part II.
A Vafa-Intriligator formula for semi-positive quotients of linear spaces
本文针对形如$V/\!/G$的光滑射影目标，建立了拟映射不变量与极大子环面商的不变量之间的联系，证明了Vafa-Intriligator公式，将生成级数表示为有限显式贡献之和。
We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mariño--Vafa type [17, 28] and GUE correlators, and the conjecture proposed in [7] that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy.
The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces
本文证明了射影曲面上层模空间虚拟 χ_y-亏格生成级数的爆破公式，统一处理 Gieseker 稳定层与 P^2 上带框架层两种情形，并基于 Nakajima-Yoshioka 爆破算法及 Kuhn-Tanaka 的推广完成证明。
We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.
Gopakumar-Vafa Invariants and Macdonald Formula
研究戈帕库马尔 - 瓦法不变量与麦克唐纳公式，通过导出可构造周环指数构建关系，在局部\(\mathbb{P}^2\)等情形下进行分析，确定相关空间、可约和项等，还在特定次数下确定多种元素并构建比较。
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection $π:\Pbb(\Escr)\rightarrow B$, for $B$ a non-singular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi,Vafa, and Witten to compute the Euler characteristic of elliptically fibered Calabi-Yau fourfolds used for F-theory compactifications of string vacua. The utility of such a formula is illustrated through applications, such as the ability to compute the Chern numbers of any non-singular complet…
We observe a general structure theorem for quantum cohomology rings, a non-homogeneous version of the usual cohomology ring encoding information about (almost holomorphic) rational curves. An application is the rigorous computation of the quantum cohomology of Grassmannians. As purely algebraic consequence we prove a beautiful formula of Vafa and Intriligator for intersection numbers of certain compactifications of moduli spaces of maps from a Riemann surface (any genus) to G(k,n) which recently has excited many mathematicians. The formula generalizes to any Fano manifold whose cohomology ring…
Gopakumar-Vafa invariants and Macdonald formula II
本文证明了局部平面和二次曲面上所有有效曲线类及欧拉特征数的上同调GV/PT对应，通过点修正交换关系排除其他支撑，并应用族Macdonald公式在Chow簇上建立恒等式。
Vafa-Witten invariants from wall-crossing for framed sheaves
本文研究光滑项目曲面的非零全纯2形式的精炼SU(r) Vafa-Witten配分函数，通过框架sheaves模空间的χ_y生成函数和新的稳定/共稳定墙穿越公式，证明了Vafa-Witten不变量的约束。
Degenerate contributions to higher genus Gromov-Witten invariants of Calabi-Yau 3-folds are computed via Hodge integrals. The vanishing of contributions of covers of elliptic curves conjectured by Gopakumar and Vafa is proven. A formula for degree 1 covers for all genus pairs is computed in agreement with M-theoretic calculations of Gopakumar and Vafa. Finally, these results lead to a proof of a formula in the tautological ring of the moduli space of curves previously conjectured by Faber.
Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these integrals from the standard descendent potential (for any target X). We use virtual localization and classical degeneracy calculations to find trigonometric closed form solutions for special Hodge integrals over the moduli space of pointed curves. These formulas are applied to two computations in Gromov-Witten theory for Calabi-Yau 3-folds. The genus g, degree d multiple cover contribution of a rational cu…
Topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the Circle Method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of Circle Method for a mock modular form of a higher depth.
Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity
本文针对满足特定条件的光滑曲面提出精细化 SU(3) Vafa-Witten 不变量猜想公式，修正既有方案并证明其满足精细化 S-对偶模性，通过模空间虚拟亏格计算与嵌套 Hilbert 概型结果提供证据。
For the resolved conifold with one outer D-brane in arbitrary framing, we present some results for the open string partition functions obtained by some operator manipulations. We prove some conjectures by Aganagic-Vafa and Aganagic-Klemm-Vafa that relates such invariants to the mirror curve of the resolved conifold. This establishes local mirror symmetry for the resolved confolds for holomorphic disc invariants. We also verify an integrality conjecture of such invariants by Ooguri-Vafa in this case and present closed formulas for some Ooguri-Vafa type invariants in genus 0 and arbitrary genera…
Virtual refinements of the Vafa-Witten formula
本文提出 Vafa-Witten 公式的虚拟精细化猜想，将秩 2 层模空间的虚拟 $χ_y$-亏格用 descendent Donaldson 不变量和 Seiberg-Witten 不变量表示，并通过局部化计算验证该猜想。
Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
本文利用约化Gromov-Witten理论为全纯辛4-流形定义了新的Gopakumar-Vafa型不变量，猜想其整数性并给出层论解释，在多个例子中验证猜想，并得到K3^{[2]}型4-流形上亏格2曲线计数的猜想公式及Fujiki常数的新闭公式。
This paper discusses the overlap of the Hori-Vafa formulation of mirror symmetry with some other constructions. We focus on compact Calabi-Yau hypersurfaces \mathcal{M}_G = {G = 0} in weighted complex projective spaces. The Hori-Vafa formalism relates a family {\mathcal{M}_G \in WCP^{m-1}_{Q_1,...,Q_m}[s] | \sum_{i=1}^m Q_i = s} of such hypersurfaces to a single Landau-Ginzburg mirror theory. A technique suggested by Hori and Vafa allows the Picard-Fuchs equations satisfied by the corresponding mirror periods to be determined. Some examples in which the variety \mathcal{M}_G is crepantly resol…
On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$
本文用拓扑顶点形式推导局部$\mathbb{P}^2$的Gromov--Witten不变式领先项公式，并应用它推广Guo--Zhou公式及证明关于$\mathbb{P}^2$上一维层模空间领先Betti数的猜想。
Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri-Vafa invariants of toric Calabi-Yau 3-folds and cohomologies of Nakajima quiver varieties. In this short note, we provide a toy model to explain this correspondence. More precisely, we study the topological open string model of $\mathbb{C}^3$ with one Aganagic-Vafa brane $\mathcal{D}_τ$, and we show that, when $τ\leq 0$, its Ooguri-Vafa invariants are given by the Betti numbers of certain quiver variety. Moreover, the existence of Ooguri-Vafa invar…
Vafa-Witten invariants from modular anomaly
本文利用模反常的普适公式，提取了Hirzebruch和del Pezzo曲面上U(N) Vafa-Witten理论的全纯生成函数，推导出所有N的瞬子模空间Betti数闭式公式，并揭示了fiber-base对偶及新不变量。
Let $X$ be a Calabi-Yau 4-fold and $D$ a smooth divisor on it. We consider tautological complex associated with $L=\mathcal{O}_X(D)$ on the moduli space of Le Potier stable pairs and define its counting invariant by integrating the Euler class against the virtual class. We conjecture a formula for their generating series expressed using genus zero Gopakumar-Vafa invariants of $D$ and genus one Gopakumar-Vafa type invariants of $X$, which we verify in several examples. When $X$ is the local resolved conifold, our conjecture reproduces a conjectural formula of Cao-Kool-Monavari in the PT chamber…
Rank $N$ Vafa-Witten invariants, modularity and blow-up
本文推导了任意秩 $N$ 下 $\mathbb{P}^2$ 的精细 Vafa-Witten 不变量生成函数及其非全纯模完备化的显式表达式，并将相关生成函数推广至一般腔，同时给出了显式模形式的吹胀公式。
$\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
本文基于S-对偶性和虚拟退化轨迹理论，提出SU(r) Vafa-Witten配分函数的结构公式，并利用theta函数、Seiberg-Witten不变量及Ramanujan连分数给出r=4,5的新表达式，同时预测稳定层模空间的虚拟Euler特征。
We derive a closed formula for the generating function of genus two Gromov-Witten invariants of quintic 3-folds and verify the corresponding mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa.
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