Witten-Kontsevich Tau-函数的显式公式

广义Kontsevich图与r-自旋相交数的拓扑递归

Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers

本文定义四类广义Kontsevich图并证明主要类型ciliated maps满足Tutte递归，从而可由拓扑递归计算，进而建立其与r-自旋相交数的恒等式，为组合理解Witten类及探索拓扑递归的辛不变性提供新工具。

Approximating tau-functions by theta-functions

We prove that the logarithm of an arbitrary tau-function of the KdV hierarchy can be approximated, in the topology of graded formal series by the logarithmic expansions of hyperelliptic theta-functions of finite genus, up to at most quadratic terms. As an example we consider theta-functional approximations of the Witten--Kontsevich tau-function.

Gromov-Witten invariants of $\mathbb{P}^1$ coupled to a KdV tau function

We consider the pull-back of a natural sequence of cohomology classes $Θ_{g,n}\in H^{2(2g-2+n)}(\overline{\cal M}_{g,n})$ to the moduli space of stable maps ${\cal M}^g_n(\mathbb{P}^1,d)$. These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via $Z^{BGW}(\hbar,t_0,t_1,...)=\exp\sum\frac{\hbar^{2g-2}}{n!}\int_{\overline{\cal M}_{g,n}}Θ_{g,n}\cdot\prod_{j=1}^nψ_j^{k_j}\prod t_{k_j}$. Insertions of the pull-backs of the classes $Θ_{g,n}$ into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target $\mathbb{P}^1$ a…

轨道褶皱的Gromov-Witten不变量讲座

Lectures on Gromov-Witten invariants of orbifolds

这是一个法医的课堂笔记2005年6月6日至11日，我在Cetraro上了一堂课。该理论描述的是陈阮的Gromov-Witten理论的orbifolds开发的Graber，Vistoli和我在代数设置，但与介绍开始在Kontsevich公式的合理平面曲线和通过Gromov-Witten理论的代数流形。由于这不是一个联合文件，我终于得到直接信贷维多利的美丽的想法！在附录中，我在得到允许的情况下，附上了孔采维奇1996年7月写给鲍里索夫的两封有关信件。研究动机整合的人可能想阅读和理解一些他们发现令人困惑的东西。

Q-Polynomial expansion for Brezin-Gross-Witten tau-function

In this paper, we prove a conjecture of Alexandrov that the generalized Brezin-Gross-Witten tau-functions are hypergeometric tau functions of BKP hierarchy after re-scaling. In particular, this shows that the original BGW tau-function, which has enumerative geometric interpretations, can be represented as a linear combination of Schur Q-polynomials with simple coefficients.

费曼图和KdV层次

Feynman diagrams and the KdV hierarchy

曲线的模空间上的稳定上同调类的相交数的生成级数满足弦方程和KdV族。Kontsevich对这一结果的原始证明使用了矩阵模型和矩阵艾里方程。威滕然后改写孔采维奇的结果在Virasoro代数，通过巧妙的混合费曼图技术和积分的部分与一些“相当可怕的选择”的被积函数。在这份说明中，我们将展示如何威滕的强大的选择可以交换标准的费曼图操作，只要一个适当扩大类的费曼图发生在证明。

Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy

In this paper, we prove that the generalized Brézin-Gross-Witten tau-function is a hypergeometric solution of the BKP hierarchy with simple weight generating function. We claim that it describes a spin version of the strictly monotone Hurwitz numbers. A family of the hypergeometric tau-functions of the BKP hierarchy, corresponding to the rational weight generating functions, is investigated. In particular, the cut-and-join operators are constructed, and the explicit description of the BKP Sato Grassmannian points is derived. Representatives of this family can be associated with interesting fam…

Gromov-Witten理论，Hurwitz数，矩阵模型，I

Gromov-Witten theory, Hurwitz numbers, and Matrix models, I

本文的主要目的是提出一种新的方法，通过Hurwitz数Kontsevich的组合/矩阵模型的相交理论的模空间的曲线。第二个目标是提出一个博览会圈的想法涉及：赫维茨号码，格罗莫夫-威滕理论的投影线，矩阵积分，和理论的随机树。更多的问题将在续集中讨论。

Buryak-Okounkov formula for the $n$-point function and a new proof of the Witten conjecture

We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new proof of the famous Witten conjecture / Kontsevich theorem, where the link between the intersection theory of the moduli spaces and integrable systems is established via the geometry of double ramification cycles.

Connecting Hodge integrals to Gromov-Witten invariants by Virasoro operators

In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety $X$ can be connected to the generating function for Gromov-Witten invariants of $X$ by a series of differential operators $\{ L_m \mid m \geq 1 \}$ after a suitable change of variables. These operators satisfy the Virasoro bracket relation and can be seen as a generalization of the Virasoro operators appeared in the Virasoro constraints for Kontsevich-Witten tau-function in the point case. This result is an extension of the work in \cite{LW} for t…

KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model

In this paper we introduce a new family of the KP tau-functions. This family can be described by a deformation of the generalized Kontsevich matrix model. We prove that the simplest representative of this family describes a generating function of the cubic Hodge integrals satisfying the Calabi-Yau condition, and claim that the whole family describes its generalization for the higher spin cases. To investigate this family we construct a new description of the Sato Grassmannian in terms of a canonical pair of the Kac-Schwarz operators.

Seiberg-Witten Theory and Topological Recursion

Kontsevich-Soibelman (2017) reformulated Eynard-Orantin topological recursion (2007) in terms of Airy structure which provides some geometrical insights into the relationship between the moduli space of curves and topological recursion. In this work, we investigate the analytical approach to this relationship using the Seiberg-Witten family of curves as the main example. In particular, we are going to show that the formula computing the Hitchin systems' Special Kahler's prepotential from the genus zero part of topological recursion as obtained by Baraglia-Huang (2017) can be generalized for a …

一种研究$G$-等变双有理不变量的Gromov-Witten方法

A Gromov-Witten approach to $G$-equivariant birational invariants

该研究依托21世纪初的几何论证建立Chen-Ruan上同调与若干$G$-双有理不变量的关联并给出应用，结合原子理论将推动等变原子理论构建，是双有理与光滑不变量关联研究计划的延伸。

Gromov-Witten不变量的热带胶合公式

Tropical gluing formulae for Gromov-Witten invariants

本文证明了两个关于爆炸流形的Gromov-Witten不变量的热带胶合公式，该公式对于利用法线交叉退化计算辛流形的Gromov-Witten不变量是有用的.第一个公式推广了Gromov-Witten不变量的辛和公式。第二个公式更强，也推广了Kontsevich和Manin的分裂和亏格归约公理。两个热带胶合公式都有包含引力子项的版本。

The ordered exponential representation of GKM using the $W_{1+\infty}$ operator

The generalized Kontsevich model (GKM) is a one-matrix model with arbitrary potential. Its partition function belongs to the KP hierarchy. When the potential is monomial, it is an $r$-reduced tau-function that governs the $r$-spin intersection numbers. In this paper, we present an ordered exponential representation of monomial GKM in terms of the $W_{1+\infty}$ operators that preserves the KP integrability. In fact, this representation is naturally the solution of a $W_{1+\infty}$ constraint that uniquely determines the tau-function. Furthermore, we show that, for the cases of Kontsevich-Witte…

Kontsevich公式与热带几何中的WDVV方程

Kontsevich's formula and the WDVV equations in tropical geometry

利用Gromov-Witten理论，通过3d-1个一般给定点的d次复平面有理曲线的个数可以用由所谓WDVV方程导出的Kontsevich公式递归计算。在本文中，我们建立了相同的结果完全在热带几何的语言。特别是这表明如何模空间的概念稳定的曲线和地图，（评价和健忘）态射，交重数和他们的不变性下变形可以结转到热带世界。

超体积与KdV tau函数

Super volumes and KdV tau functions

本文通过引入Ramond穿孔，证明超曲线模空间的超Weil-Petersson体积与广义BGW KdV tau函数相关，并满足Stanford-Witten递归。

GUE相关子项在KdV可积性中的可积性

KdV integrability in GUE correlators

本文基于GUE相关子项与Witten交集数的关系，通过Toda晶格的tau函数证明了Witten-Kontsevich定理，揭示了KdV可积结构与GUE相关子项的联系。

Hodge and Prym tau functions, Jenkins-Strebel differentials and combinatorial model of $\mathcal M_{g,n}$

The principal goal of the paper is to apply the approach inspired by the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Jenkins-Strebel differentials. The line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $\mathcal L_j$ and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials whe…

Hirota Quadratic Equations for the Gromov--Witten Invariants of $\mathbb{P}_{n-2,2,2}^1$

Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case the Kac--Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac--Wakimoto hierarchy.

Kontsevich-Hurwitz配分函数的维拉索罗约束

Virasoro constraints for Kontsevich-Hurwitz partition function

M.Kazarian和S.Lando在Kontsevich和Hurwitz配分函数之间找到了一个1-参数插值，它完全位于KP τ-函数的空间内。V.Bouchard和M.Marino提出这种插值满足某些变形的维拉索罗约束。然而，他们描述的限制，在一个有点复杂的形式的AMM-埃纳德方程，而涉及朗伯光谱曲线。在这里，我们提出了相关的家庭维拉索罗约束明确。它们与传统的连续维拉索罗约束的不同之处在于最简单的可能方式：L_{-1}算子的恒定位移u^2/24，其中u是Kontsevich和Hurwitz模型之间的插值参数。弦方程的这种微不足道的修改产生了整个变形，它是维拉索罗约束L_m -> U L_m U^{-1}的共轭，并“扭曲”了配分函数，Z_{KH} = U Z_K。共轭U是通过以前未被注意到的算子来表示的，这些算子使Kontsevich模型的准经典（平面）自由能零化，但不属于通用格拉斯曼的对称群GL（\infty）。

Tau-structure for the Double Ramification Hierarchies

In this paper we continue the study of the double ramification hierarchy of [Bur15]. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from [Bur15]: for any semisimple cohomological field theory, the Dubrovin-Zhang and double ramification hierarchies are related by a normal (i.e. preserving the tau-structure [DLYZ…

栈的亏格为零的Gromov-Witten不变量的重构定理

A reconstruction theorem for genus zero Gromov-Witten invariants of stacks

本文从两个方面推广了Kontsevich和Manin的第一重构定理。首先，我们允许目标空间是Deligne-Mumford栈。其次，在一定的收敛性假设下，证明了不需要在上同调环上，而只需在小量子上同调环族中的任意量子环上检验$H^2$ -生成的假设。

Topological recursion with hard edges

We prove a Givental type decomposition for partition functions that arise out of topological recursion applied to spectral curves. Copies of the Konstevich-Witten KdV tau function arise out of regular spectral curves and copies of the Brezin-Gross-Witten KdV tau function arise out of irregular spectral curves. We present the example of this decomposition for the matrix model with two hard edges and spectral curve $(x^2-4)y^2=1$

A generalization of Witten's conjecture for the Pixton class and the noncommutative KdV hierarchy

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative KdV hierarchy, which we introduced in a previous work. The specialization of this conjecture to the top degree part of Pixton's class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we …

带有碰撞标记的单纯稳定曲线的Virasoro递推关系

Virasoro Recursions for Simplicially Stable Curves with Colliding Markings

本文针对带有碰撞标记的单纯稳定曲线，通过分解约化态射得到重构公式，推导修正后的Virasoro递推关系，并将有限支撑理论封装于Fock空间，拓展了Witten-Kontsevich势的相关研究。

Intersection numbers on $\overline {\mathcal M}_{g,n}$ and BKP hierarchy

In their recent inspiring paper Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar conjecture for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

The quantum tropical vertex

Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the $q$-refined 2-dimensional Kontsevich-Soibelman scattering diagrams compute, after the change of variables $q=e^{i \hbar}$, generating series of certain higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces. This result provides a mathematically rigorous realization of the physical derivation of the refined wall-crossing formula from topological string theory proposed by Cecotti-…

量子相交数与$\boldsymbol{\rm \text{CP}^1}$的Gromov-Witten不变量

Quantum intersection numbers and the Gromov-Witten invariants of $\mathbb{CP}^1$

本文将量子相交数与带Hodge类插入的$\rm \text{CP}^1$稳态相对Gromov-Witten不变量关联，用无限楔形式方法给出量子相交数纯量子部分显式公式的简短证明，建立其与单分双Hurwitz数的联系。

Tau函数与微分模

Tau function and moduli of differentials

复代数曲线上一般全纯1-微分模空间上的τ函数被解释为稳定曲线的模空间上的投影Hodge丛上的线丛的一部分。计算了τ函数在1-微分的模空间边界附近的渐近性，并给出了投影化Hodge丛上Hodge类的拉回用重言式类和边界因子类表示的显式表达式.这个表达式被用来阐明与Hodge丛上的Teichmüller流相关的李雅普诺夫指数之和的Kontsevich-Zorich公式的几何意义。

Gelfand-Dikii族的Witten解

Witten solution of the Gelfand-Dikii hierarchy

在n-Gelfand-Dikii族的解中，存在一个满足弦方程的显著解W.我们称之为威滕解，因为根据威滕猜想，函数F（x_1，x_2，x_3，.）= W（x_1，（x_2）/2，（x_3）/3，.）是n-自旋Riemann曲面的模空间的Mumford-Morita-Muller上同调类的交数的母函数。这一猜想证明了孔采维奇为n=2和威滕自己的表面属0。本文给出了W.这就把Witten猜想简化为Mamford-Morita-Muller数满足相同关系的猜想。在Witten猜想成立的条件下，这些关系也给出了计算$n$ -spin Mamford-Morita-Muller数的算法.并且证明了F（x_1，x_2，.，x_{n-1}，0，0，.）= W（x_1，（x_2）/2，.，（x_{n-1}）/（n-1），0，0，.）。

Kontsevich公式：从经典与量子视角看有理曲线

Kontsevich's Formula for Rational Curves from Classical and Quantum Perspectives

本文从经典组合与量子上同调两个角度给出Kontsevich有理曲线计数公式的证明，并推广至双有理曲线情形。

Witten的D_4可积族猜想

Witten's D_4 Integrable Hierarchies Conjecture

本文证明了Fan-Jarvis-Ruan-Witten理论对具有对称群的D_4<J>和具有对称群G_{max}的D_4 ^T的总后裔势函数都是D_4 Kac-Wakimoto/Drinfeld-Sokolov族的τ-函数。这完成了证明，开始在[FJR]，威滕可积的层次猜想的所有简单的（ADE）奇点。

A_{n-1}奇点与nKdV族

A_{n-1} singularities and nKdV hierarchies

根据E. Witten证明了M. Kontsevich，Riemann曲面的Deligne-Mumford模空间上的交指标的生成函数与KdV族的τ-函数重合。在辛流形的Gromov-Witten不变量理论中，生成函数被自然地推广为{ \em全下降势}。论文arXiv：AG/0108100和arXive：math.DG/0108160包含两个等价的构造，它们是由Gromov-Witten理论中的一些结果激发的，这些结果将全下降势与任何半单Frobenius结构相关联。本文证明了在$A_{n-1}$ -奇异性的极小变形上的K.Saito的Frobenius结构的情况下，总下降势是$n$ KdV族的τ-函数.我们得出这个结果从一个更一般的建设解决方案的$n$ KdV层次从$n-1$解决方案的KdV层次。

引力耦合拓扑西格玛模型的关联函数之间的关系

Relations between the correlators of the topological sigma-model coupled to gravity

我们证明了一个新的递归关系之间的关联函数$< τ_{d_1}γ_1...τ_{d_n}γ_n >_{g,β}$，它与已知的关系一起允许一个表达所有这些通过完整的系统的Gromov-Witten不变量的意义上的Kontsevich-Manin和交叉指数的重言式类$\bar{M}_{g,n}$，有效地计算鉴于早期的结果，由于芒福德，Kontsevich，Getzler和费伯。这种关系表明，大相空间坐标的线性变化将带有引力子的势变换为另一个完全根据Gromov-Witten对应和$V^n\times\bar{M}_{g,n}$上的相交理论定义的函数。然后，我们将引力后裔的形式主义从量子上同调推广到更一般的Frobenius流形和上同调场论。

分解锥状结构的Gromov-Witten不变量的费米表示

On fermionic representation of the Gromov-Witten invariants of the resolved Conifold

我们证明，费米形式的生成函数的Gromov-Witten不变量的解决conifold是一个Bogoliubov变换的费米真空，特别是，它是一个tau函数的KP层次。我们的证明是基于Aganagic等人提出并在我们最近的工作中证明的拓扑顶点的胶合规则和框架单腿和双腿拓扑顶点的费米子表示公式。

曲线模空间上相交数的n点函数

The n-point functions for intersection numbers on moduli spaces of curves

利用著名的Witten-Kontsevich定理，证明了曲线模空间上相交数的$n$点函数的一个递推公式。它已被用来证明Faber相交数猜想，并激励我们找到一些Gromov-Witten不变量的几何消失恒等式。后者最近已被X. Liu和R. Pandharipande我们还给出了一个组合的解释$n$ -点函数的总和在二叉树。

$\Mbar_{1,4}$与椭圆Gromov-Witten不变量的交理论

Intersection theory on $\Mbar_{1,4}$ and elliptic Gromov-Witten invariants

WDVV方程可以由任何二维拓扑场论的亏格0关联函数满足，并且可以用来确定许多射影簇的亏格0（有理）Gromov-Witten不变量（就像Kontsevich对射影空间所做的那样）。在本文中，我们提出了一个方程的一个类似的普遍性质的亏格1（椭圆）Gromov-Witten不变量-然而，它是更复杂的比WDVV方程，它的几何意义是不清楚我们。(Our证明是间接的）。然而，我们表明，这个方程足以确定椭圆Gromov-Witten不变量的射影空间。在本文的续篇中，我们将证明这个方程是唯一一个不同于WDVV方程的方程，它涉及二维拓扑场论与拓扑引力耦合的椭圆和有理关联函数。目前还不清楚在高等亏格的小相空间上是否存在这类方程，但我们认为这是不可能的。(The亏格0和亏格1的情况是特殊的，因为小相空间上的关联函数确定大相空间上的那些。

曲线模空间上的组合与代数几何上同调类

Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves

基于Strebel微分对曲线的模空间的组合描述，Witten和Kontsevich引入了组合上同调类$W_{(m_0,m_1,m_2,\dots),n}$，并证明了它们可以用Mumford-Morita-Miller类来表示.有人认为，这种联系应提供的一个定理迪弗朗切斯科，Itzykson和Zuber有关的衍生物的维滕-Kontsevich配分函数相对于一组变量的衍生物相对于其他一组变量。有两件事显示。首先，这在复余维1中起作用。其次，在所有的情况下，当它已经有可能使迪弗朗切斯科，Itzykson和祖伯correspondence明确这转化为身份的类型$$ \int_{W_{(m_0,m_1,m_2,\dots),n}}\prodψ_i^{d_i} =\int_{\overline{\cal{M}}_{g,n}} X_{(m_0,m_1,m_2,\dots),n}\prodψ_i^{d_i} $$其中的$X_{(m_0,m_1,m_2,\dots),n}$是明确的多项式在代数几何类和$ψ_i$是陈类的点丛，任何选择的$d_1,\dots,d_n$。

射影直线上的等变orbifold结构与可积族

Equivariant orbifold structures on the projective line and integrable hierarchies

设$\CP^1_{k,m}$是$\CP^1$上的轨道结构，它是通过分别用$z\mapsto z^k$和$w\mapsto w^m.$对0和$\infty$的邻域进行一致化而得到的。环面对射影线的对角作用自然地导出了对$\CP^1_{k,m}.$的轨道作用。本文证明了：如果k和m互质，则Givental对等变全下降Gromov的预测-$\CP^1_{k,m}$的Witten势满足一定的Hirota二次方程（简称HQE）。我们还表明，经过适当的变化的变量，类似于Getzler的变化的等变Gromov-Witten理论的$\CP^1$，HQE变成HQE的2-Toda的层次，即，$\CP^1_{k,m}$的Gromov-Witten势是2-Toda体系的τ函数。更准确地说，我们得到了一个序列的τ-函数的2-Toda的层次从下降的潜力，通过一些翻译。后一个条件，即序列中的所有tau函数都是通过平移从单个函数中获得的，这对2-Toda层次的解施加了严重的约束。我们的定理导致发现一个新的可积族（我们建议被称为等变双分次Toda族）。我们推测，这一新的等级制度，即，唯一地确定了$\CP^1_{k,m}.$的等变Gromov-Witten不变量

Donaldson-Witten理论中的Blowup公式与可积族

Blowup formulae in Donaldson-Witten theory and integrable hierarchies

利用超椭圆Kleinian函数理论研究了规范群为SU（N）的Donaldson-Witten理论的爆破公式。我们发现，爆破功能是一个超椭圆σ函数，我们描述了一个明确的程序来扩大它的规范组的Casimirs到任意阶。作为一个推论，我们得到了一个新的表达式的接触条款，我们表明，关联函数涉及的例外因子是由KdV层次。我们还表明，对于简单型流形，爆破函数成为一个多孤子解的τ函数。

Hamilton发展偏微分方程的Hodge积分和τ-对称可积族

Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

对于任意半单Frobenius流形，我们构造了Hamilton偏微分方程的Hodge可积族.在量子上同调的特殊情况下，解的τ-函数产生了Gromov-Witten类及其后代的相交数，以及稳定映射的模空间上的霍奇丛的特征类。对于一维Frobenius流形的霍奇层次是一个变形的Korteweg-de Vries层次取决于一个无限数量的参数。猜想这个族是具有τ-函数的纯量Hamilton可积族中的一个泛对象。

N = 2超对称规范理论中的Whitham变形和Tau函数

Whitham Deformations and Tau Functions in N = 2 Supersymmetric Gauge Theories

我们回顾了最近在N=2超对称规范理论及其拓扑扭曲版本中发现的可积系统的新方面。主要议题是（i）经典规范群的Seiberg-Witten曲线的Whitham变形的显式构造，（ii）它在拓扑扭曲理论的u-平面积分中的接触项的应用，以及（iii）拓扑扭曲理论中tau函数和爆破公式之间的联系。