In Part 1 of this paper, we study gravitational descendents of Gromov-Witten invariants for general projective manifolds, applying the Behrend-Fantechi construction of the virtual fundamental classes.
In Part 2, we calculate the topological recursion relations in genus 2. There are two of these, one for the second descendent of a field, and one for the first descendents of two fields. The proof uses the results of Part 1 together with a thorough study of intersection theory on the moduli space $\bar{M}_{2,2}$.
We construct topological recursion relations (TRR's) at higher genera $g\ge2$ for general 2-dimensional topological field theories coupled to gravity. These TRR's when combined with Virasoro conditions enable one to determine the number of higher genus holomorphic curves in any Fano varieties. In the case of $CP^2$ we reproduce the known results at genus $g=2$.
In this paper, we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton's relations on the moduli space of curves. As an application, we prove Pixton's relations imply a known topological recursion relation on $\bar{\mathcal{M}}_{g,1}$ for genus $g\leq4$.
New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold $X$ are exhibited. When the cohomology of $X$ is generated by divisor classes and classes ``with vanishing one-point invariants,'' the relations determine many-point invariants in terms of one-point invariants.
We state and prove a topological recursion relation that expresses any genus-g Gromov-Witten invariant of a projective manifold with at least a (3g-1)-st power of a cotangent line class in terms of invariants with fewer cotangent line classes. For projective spaces, we prove that these relations together with the Virasoro conditions are sufficient to calculate the full Gromov-Witten potential. This gives the first computationally feasible way to determine the higher genus Gromov-Witten invariants of projective spaces.
According to \cite{BOSS} and \cite{M1}, the ancestor correlators of any semi-simple cohomological field theory satisfy {\em local} Eynard--Orantin recursion. In this paper, we prove that for simple singularities, the local recursion can be extended to a global one. The spectral curve of the global recursion is an interesting family of Riemann surfaces defined by the invariant polynomials of the corresponding Weyl group. We also prove that for genus 0 and 1, the free energies introduced in \cite{EO} coincide up to some constant factors with respectively the genus 0 and 1 primary potentials of t…
Let $X$ be a smooth projective surface and $D$ a smooth rational ample divisor in $X$. We prove an all-genus generalization of the genus $0$ WDVV equation for primary Gromov--Witten invariants of the local 3-fold $\mathcal{O}_X(-D)$. The proof relies on a correspondence between all-genus Gromov--Witten invariants and refined Donaldson--Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations.
A recursion relation of hyperelliptic psi functions of genus two, which was derived by D.G. Cantor (J. reine angew. Math. 447 (1994) 91-145), is studied. As Cantor's approach is algebraic, another derivation is presented as a natural extension of the analytic derivation of the recursion relation of the elliptic psi function.
In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves.
We describe Zhu recursion for a vertex operator algebra (VOA) on a general genus Riemann surface in the Schottky uniformization where $n$-point correlation functions are written as linear combinations of $(n-1)$-point functions with universal coefficients. These coefficients are identified with specific geometric structures on the Riemann surface. We apply Zhu recursion to the Heisenberg VOA and determine all its correlation functions. For a general VOA, Zhu recursion with respect to the Virasoro vector is shown to lead to conformal Ward identities expressed in terms of derivatives with respec…
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.
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
In this paper, we give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying new relations in the tautological ring of the moduli space of 2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.
We derive a Bouchard--Eynard type topological recursion for the total descendant potential of $A_N$-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of $A_N$-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of $\mathcal{W}$-algebra constraints.
We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these numbers are useful for computing genus-one enumerative invariants of projective spaces and Gromov-Witten invariants of complete intersections.
Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective line. As a consequence, nontrivial tautological classes in the kernel of the push-forward map associated to the irreducible boundary divisor of the moduli space of stable g+1 curves are constructed. The geometry of genus g+1 curves then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities con…
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.
We calculate the genus 2 correlation functions of two-dimensional topological gravity, in a background with two primary fields, using the genus 2 topological recursion relations.
The string corrections of tree-level open-string amplitudes can be described by Selberg integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. This allows for a recursion of the $α'$-expansion of tree-level string corrections in the number of external states using the Drinfeld associator. While the feasibility of this recursion is well-known, we provide a mathematical description in terms of twisted de Rham theory and intersection numbers of twisted forms. In particular, this leads to purely combinatorial expressions for the matrix representation of the Lie algebra generators appearing i…
We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers. The main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine, "Chiodo formulas for the r-th roots and topological recursion", Lett. Math. Phys. (2016).
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 …
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…
We present a higher genus generalization of $bc$-Motzkin numbers, which are themselves a generalization of Catalan numbers, and we derive a recursive formula which can be used to calculate them. Further, we show that this leads to a topological recursion which is identical to the topological recursion that had previously been proved by Dumitrescu and Mulase for generalized Catalan numbers, and which is an example of the Eynard-Orantin topological recursion.
We use the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{disc}$ (discrete volumes), to express Gaussian means in all genera as polynomials in special times weighted by ancestor invariants of an underlying cohomological field theory. We translate topological recursion of the Gaussian model into recurrent relations for coefficients of this expansion proving their integrality and positivit…
We study the constant contributions to the free energies obtained through the topological recursion applied to the complex curves mirror to toric Calabi-Yau threefolds. We show that the recursion reproduces precisely the corresponding Gromov-Witten invariants, which can be encoded in powers of the MacMahon function. As a result, we extend the scope of the "remodeling conjecture" to the full free energies, including the constant contributions. In the process we study how the pair of pants decomposition of the mirror curves plays an important role in the topological recursion. We also show that …
We identify the Givental formula for the ancestor formal Gromov-Witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve. As an application we prove a conjecture of Norbury and Scott on the reconstruction of the stationary sector of the Gromov-Witten potential of $\CP1$ via a particular spectral curve.
GW/DT invariants and 5D BPS indices for strips from topological recursion
IHES（高等科学研究所）；University of Geneva（日内瓦大学）；Université Jean Monnet（让·莫内大学）；Institut Camille Jordan UMR 5208（卡米耶·若尔当研究所）；Institut Universitaire de France（法兰西学院）
The purpose of this paper is to give a twisted version of the Eynard-Orantin topological recursion by a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic definition for a topological recursion to define how to twist a standard topological recursion by a 2D TQFT. The A-model side enumerative problem consists of counting cell graphs where in addition vertices are decorated by elements in a Frobenius algebra, and which are a twisted version of the generalized Catalan numbers of Dumitrescu-Mulase-Safnuk-Sorkin. We show that the function which counts these d…
We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions o…
We prove that the topological recursion reconstructs the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves). This includes nearly all previously known cases in the literature, and many more; in particular, it includes many quantum curves of order greater than two. We also explore the connection between the choice of ordering in the quantization of the spectral curve and the choice of integration divisor to reconstruct the WKB expansion.
On a stack of stable maps, the psi classes are modified by subtracting certain boundary divisors. These modified psi classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow simple expressions in terms of modified psi classes. Topological recursion relations are established among their top products in genus zero, yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety. In higher genus, the obtained numbers are only virtual, due to contributions from spurious components of th…
We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two recursions, namely the string equation and the dilaton equation. We establish, for each integer m>0, an analogous pair of recursions which determine these intersection numbers on the moduli spaces of m-stable pointed curves of genus one.
In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin …
In this paper we investigate the algebraic structure related to a new type of correlator associated to the moduli spaces of $S^1$-parametrized curves in contact homology and rational symplectic field theory. Such correlators are the natural generalization of the non-equivariant linearized contact homology differential (after Bourgeois-Oancea) and give rise to an invariant Nijenhuis (or hereditary) operator (à la Magri-Fuchssteiner) in contact homology which recovers the descendant theory from the primaries. We also sketch how such structure generalizes to the full SFT Poisson homology algebra …
Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces. In t…
Any topological recursion on a rational spectral curve is KP integrable
Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究院几何物理中心）；University of Haifa（海法大学）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术学院）；NRC “Kurchatov Institute” – ITEP（库尔恰托夫研究所国家研究中心-理论物理与实验物理研究所）；Igor Krichever Center for Advanced Studies, Skoltech（斯科尔科沃伊戈尔克里切弗高级研究中心）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schrödinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way. The Schrödinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed in a rather different way via topological recursion.
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. …
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.…
We analyze Chiodo's formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with psi-classes are reproduced via the Chekhov-Eynard-Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.
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…
In this paper we prove a mirror symmetry conjecture based on the work of Brini-Eynard-Mariño \cite{BEM} and Diaconescu-Shende-Vafa \cite{DSV}. This conjecture relates open Gromov-Witten invariants of the conifold transition of a torus knot to the topological recursion on the B-model spectral curve.
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 $(x^2-4)y^2=1$
