Lin Chen, Yi Li, Kefeng Liu
In this note, we use the method of [3] to give a simple proof of famous Witten conjecture. Combining the coefficients derived in our note and this method, we can derive more recursion formulas of Hodge integrals.
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We give a short and direct proof of the $λ_g$-Conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the ``polynomiality'' of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures.
This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several res…
In this ``experimental'' research, we use known topological recursion relations in genera-zero, -one, and -two to compute the n-point descendant Gromov-Witten invariants of P^1 for arbitrary degrees and low values of n. The results are consistent with the Virasoro conjecture and also lead to explicit computations of all Hodge integrals in these genera. We also derive new recursion relations for simple Hurwitz numbers similar to those of Graber and Pandharipande.
We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The stationary sector is formed, by definition, by the descendents of the point class. Completed cycles arise naturally in the theory of shifted symmetric functions. Using this equivalence, we give a complete description of the stationary Gromov-Witten theory of the projective line and elliptic curve. Toda equations for the relative stationary theory of the projective line are derived.
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of ideas involved: Hurwitz numbers, Gromov-Witten theory of the projective line, matrix integrals, and the theory of random trees. Further topics will be treated in a sequel.
In this letter,we present our conjecture on the connection between the Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the $GL(\infty)$ group element. An important feature of this group element is its simplicity: this is a group element of the Virasoro subalgebra of $gl(\infty)$. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the…
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permuations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for Hodge integrals on the moduli space of curves, has been a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one "lambda-class"), and show that after a non-trivial change of variables, G=H in positive genus, where H is a generating series for Hurwitz numbers.…
Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula expressing Hurwitz numbers (counting covers of the projective line with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebro-geometric definition of a "relative space".
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2-sphere.
We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Guéré and Rossi in \cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $ε$ and a quantum parameter $\hbar$. When $\hbar=0$, this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves. We establish a link between the $ε=0$ part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number non-equivalent h…
Gromov-Witten/Hurwitz wall-crossing
本文提出 $\varepsilon$-容许稳定性并证明 GW/Hurwitz 穿墙公式，进而将相对 PT/GW 对应与 Ruan 扩张 crepant 消解猜想等价化，并验证环面 del Pezzo 曲面情形。
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.
Hurwitz numbers count ramified covers of a Riemann surface with prescribed monodromy. As such, they are purely combinatorial objects. Tautological classes, on the other hand, are distinguished classes in the intersection ring of the moduli spaces of Riemann surfaces of a given genus, and are thus ``geometric.'' Localization computations in Gromov-Witten theory provide non-obvious relations between the two. This paper makes one such computation, and shows how it leads to a ``master'' relation (Theorem 0.1) that reduces the ratios of certain interesting tautological classes to the pure combinato…
Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological recursion, which is a mirror B-model corresponding to these counting problems. We show that for the case of orbifold Hurwitz numbers, the mirror objects, i.e., the spectral curve and the differential forms on it, are constructed solely from the edge-contraction operations of the counting problem in genus $0$ and one marked point. This forms a pa…
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…
The Spin Gromov-Witten/Hurwitz correspondence for $\mathbb{P}^1$
本文研究 $\mathbb{P}^1$ 的 spin Gromov-Witten 理论，利用环面作用证明等变势满足 2-BKP 层级，从而证明了 J. Lee 猜想的 GW/Hurwitz 对应的 spin 类似物，并推广至一般目标 spin 曲线。
In this expository article, we explain how to use localization to compute Gromov-Witten invariants of smooth toric varieties and orbifold Gromov-Witten invariants of smooth toric Deligne-Mumford stacks.
We propose two conjectures on Huwritz numbers with completed $(r+1)$-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard-Mariño conjecture for ordinary Hurwitz numbers. Our $r$-ELSV formula is an equality between a Hurwitz number and an integral over the space of $r$-spin structures, that is, the space of stable curves with an $r$th root of the canonical bundle. Our $r$-BM conjecture is the statement that $n$-point functions for Hurwitz numbers satisfy the topological recursion associ…
We state a version of the crepant resolution conjecture for total ancestor potentials for surface singularities, and reduce the conjecture to the quantum McKay correspondence conjecture of J.Bryan and A.Gholampour and a vanishing conjecture for Hurwitz-Hodge integrals. In particular, for singularities of type A, we prove the conjecture. We also suggest an approach towards a proof for the general cases by Teleman's reconstruction theorem for semisimple cohomological field theories.
We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino conjecture and places Hurwitz-Hodge integrals, which arise in the Gromov--Witten theory of target curves with orbifold structure, in the context of the Eynard-Orantin topological recursion.
We study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields to an interpretation in terms of tropical geometry involving local multiplicities given by Gromov-Witten invariants. Furthermore, this enables us to prove that a main result of Cavalieri-Johnson-Markwig-Ranganathan is a actually a refinement of the Gromov-Witten/Hurwitz correspondence by Okounkov-Pandharipande.
The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These "spin Hurwitz numbers", recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors' previous work, they are also related to the Gromov-Witten invariants of Kahler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kahler surfaces with positive geometric genus. The proof uses a degeneration of spin curves, an invariant defined by the spec…
We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S_n. We also find a generating function for Hodge integrals on the moduli space M_{g,2} of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.
Signed Real Hurwitz numbers
本文研究了带符号实 Hurwitz 数，通过带符号枚举实 Riemann 面间的全纯映射并将其与实 Gromov-Witten 理论相联系，同时提出了有限群主 G-丛的带符号计数以构建 Klein 拓扑量子场论。
A new spin on Hurwitz theory and ELSV via theta characteristics
本文利用 BKP tau-函数解释研究 spin Hurwitz 数，给出其多项式性质并提出由谱曲线驱动的新型拓扑递归，且证明该递归等价于含 Chiodo 类与 2-spin Witten 类的 ELSV 型公式。
We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of [C3/(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.
On the large genus of Hurwitz numbers
本文通过基本方法研究Hurwitz数在高 genus 下的渐进行为，涵盖单、双及任意固定分支类型，以及简单或包含完整循环等分支类型，总结了主要贡献。
We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by ``top intersections'' of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a ``combinatorialization'' of top intersections of psi-classes. As genus 0 double Hur…
In this paper, we study genus $0$ equivariant relative Gromov-Witten invariants of $\mathbb{P}^1$ whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighboring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber st…
We prove a conjecture of Durr, Kabanov and Okonek which provides an algebro-geometric theory of Seiberg-Witten invariants for all smooth projective surfaces. Our main technique is the cosection localization principle of virtual cycles.
We obtain a system of relations between linear Hodge integrals. As an application, we show that its first non-trivial relation implies the Witten's Conjecture/Kontsevich Theorem.
Double Hurwitz numbers count branched covers of the projective line with fixed branch points, with simple branching required over all but two points 0 and infinity, and the branching over 0 and infinity specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers have a rich structure, explored by authors in many fields. The ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves, and has led to many consequences. We determine the structure of double Hurwitz numbers using geometry, algebra, and representation theory. O…
The moduli stack of Mixed Spin P-fields (MSP) provides an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. This paper is to apply the algorithm in genus one case. We use the localization formula, the proposed algorithm in [CLLL1, CLLL2], and Zinger's packaging technique to compute the genus one Gromov-Witten invariants of quintic Calabi-Yau threefolds. New hypergeometric series identities are also discovered in the process.
Using obstruction bundles, composition law and the localization formula, we compute certain 3-point genus-0 Gromov-Witten invariants of the Hilbert scheme of 3-points on the complex projective plane. Our results partially verify Ruan's conjecture about quantum corrections for this Hilbert scheme.
Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed $\pm 1$ according to the parity of the covering surface. These numbers were first defined by Eskin-Okounkov-Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and Maulik-Pandharipande. In this paper, we construct a (spin) TQFT which computes these numbers, and deduce a formula for any genus in terms o…
Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers
本研究通过费米子计算与玻色-费米对应推导二维Toda格点层级对角tau函数的连通$(n,m)$点函数显式公式，计算其KP仿射坐标，并提出统一方法计算各类双Hurwitz数及$\mathbb P^1$的稳态Gromov-Witten不变量。
For a Landau Ginzburg space ([C^n/G],W), we construct the Witten's top Chern classes as algebraic cycles via cosection localized virtual cycles in case all sectors are narrow. We verify all axioms of such classes. We derive an explicit formula of such classes in the free case. We prove that this construction is equivalent to the prior constructions of Polishchuk-Vaintrob, of Chiodo and of Fan-Jarvis-Ruan.
We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for t…
We define a formalism for computing open orbifold GW invariants of [C^3/G] where G is any finite abelian group. We prove that this formalism and a suitable gluing algorithm can be used to compute GW invariants in all genera of any toric CY orbifold of dimension 3. We conjecture a correspondence with the DT orbifold vertex of Bryan-Cadman-Young.
Algorithms for Gromov-Witten Invariants of Elliptic Curves
提出一种基于代数与tropical几何对应的高效算法，利用Singular和OSCAR计算椭圆曲线Gromov-Witten不变量及Hurwitz数的生成级数，显著提升计算效率并促进其数论与几何性质研究。
The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of P^1. A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete and effective solution. The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of C^2, and the orbifold quantum cohomology the symmetric product of C^2. The results of the paper provi…
Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramif…
Super Gromov-Witten Invariants via torus localization
本文提出通过环面局部化定义超级 Gromov-Witten 不变量，绕开超几何相交理论，证明其满足推广的 Kontsevich-Manin 公理，并给出射影空间亏格零不变量的计算方法与显式数值。
A simple corollary of the localization theorem (due to the author and, independently, to Lian-Liu-Yau) is applied to several problems in enumerative geometry. New formulas for Schubert calculus on flag manifolds, due to Kong, and a new reconstruction theorem for genus-zero Gromov-Witten invariants, due to Bertram-Kley, are discussed, as well as some simple functorial properties of Givental's J-function. This paper will appear in the issue of the Michigan Mathematical Journal dedicated to Bill Fulton.
We give a proof of Pixton's generalized Faber-Zagier relations in the tautological Chow ring of $\overline M_{g,n}$. The strategy is very similar to the work of Pandharipande-Pixton-Zvonkine, who have given a proof of the same result in cohomology. The main tool is the Givental-Teleman classification of semisimple cohomological field theories, which, while in general only known in cohomology, via virtual localization can be shown to be also valid in Chow for the equivariant Gromov-Witten theory of the projective line. We obtain the relations just from this theory.
Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
本文证明Gaiotto向量可解析延拓至有限能量尺度，由半Seiberg--Witten谱曲线上的拓扑递归计算，并推导出与相交理论、Hurwitz数及量子曲线的联系，同时推广至更一般的Whittaker向量。
These notes were born out of a five-hour lecture series for graduate students at the May 2018 Snowbird workshop Crossing the Walls in Enumerative Geometry. After a short primer on equivariant cohomology and localization, we provide proofs of the genus-zero mirror theorems for the quintic threefold, first in Fan-Jarvis-Ruan-Witten theory and then in Gromov-Witten theory. We make no claim to originality, except in exposition, where special emphasis is placed on peeling away the standard technical machinery and viewing the mirror theorems as closed-formula manifestations of elementary localizatio…
We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with predictions from open string mirror symmetry. To this aim we set up a computation of open string invariants in the spirit of Katz-Liu, defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [C^3/Z_3], where we…
We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drasticall…
A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of $X=\mathcal{O}_{\mathbb{P}^{1}}(-1,-1)$ to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds $L_{K}\subset X$ obtained from the conormal bundles of knots $K\subset S^{3}$. This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For $(r,s)$ torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bot…
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。
We develop a gluing algorithm for Gromov-Witten invariants of toric Calabi-Yau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing.
A generalization of the Witten conjecture through spectral curve
本文提出Witten猜想的推广，通过广义拉普拉斯变换证明其几何部分，并证明单边界情形的可积性，同时推广验证了负r-spin理论的rKdV可积性及Hurwitz空间相关的KdV可积性。
We prove Iqbal's conjecture on the relationship between the free energy of closed string theory in local toric geometry and the Wess-Zumino-Witten model. This is achieved by first reformulating the calculations of the free energy by localization techniques in terms of suitable Feynman rule, then exploiting a realization of the Feynman rule by free bosons. We also use a formula of Hodge integrals conjectured by the author and proved jointly with Chiu-Chu Melissa Liu and Kefeng Liu.
Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory
研究五维\(\mathcal{N}=1\) \(U(N)\)超对称杨 - 米尔斯理论在复Kähler曲面相关空间上的配分函数，扩展前人工作，通过路径积分定域及处理极点，发现非偶数陈类贡献与精细化Vafa-Witten不变量一致，偶数陈类情况待明确。
In this paper, we propose a definition of genus one real Gromov-Witten invariants for certain symplectic manifolds with real a structure, including Calabi-Yau threefolds, and use equivariant localization to calculate certain genus one real invariants of the projective space. For this definition, we combine three moduli spaces corresponding to three possible types of involutions on a symplectic torus, by gluing them along common boundaries, to get a moduli space without codimension-one boundary and then study orientation of the total space. Modulo a technical conjectural lemma, we can prove tha…
Genus expansion of matrix models and $\hbar$ expansion of $B$KP hierarchy
本文研究矩阵模型亏格展开与$B$KP层级$\hbar$展开的联系，证明BGW模型、Kontsevich模型及spin Hurwitz数生成函数在插入$\hbar$后均为具有良好准经典行为的$\hbar$-$B$KP层级解，并揭示了模型间形变规则的相似性与差异。
We derive a fixed-point formula for integrals on moduli spaces of stable maps to projective spaces of even dimension. This gives a formula for the equivariant open Gromov-Witten invariants of (RP^{2m},CP^{2m}) and the structure constants of the equivariant Fukaya A-infinity algebra of RP^2m in CP^{2m} with bulk deformations. The formula involves contributions from Givental's correlators for the closed theory and the descendent integrals of discs, and specializes to give a new expression for the Welschinger count of real rational curves in the plane passing through some real and conjugation inv…
We describe the applications of localization methods, in particular the functorial localization formula, in the proofs of several conjectures from string theory. Functorial localization formula pushes the computations on complicated moduli spaces to simple moduli spaces. It is a key technique in the proof of the general mirror formulas, the proof of the Hori-Vafa formulas for explicit expressions of basic hypergeometric series of homogeneous manifolds, the proof of the Mariño-Vafa formula, its generalizations to two partition analogue. We will also discuss our development of the mathematical t…
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