Xiaobo Liu, Gang Tian
Eguchi-Hori-Xiong and S. Katz proposed a conjecture that the partition function of topological sigma model coupled to gravity is annihilated by infinitely many differential operators which form half branch of the Virasoro algebra. In this paper, we give a proof to this conjecture for the genus 0 part.
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We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are completely and explicitly determined.
We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds $M$ ($c_1(M)>0$) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to $χ(M)$, the Euler characteristic of the manifold $M$.
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 our previous article we have proposed that the Virasoro algebra controls the quantum cohomology of Fano varieties at all genera. In this paper we construct a free field description of Virasoro operators and quantum cohomology. We shall show that to each even (odd) homology class of a Kähler manifold we have a free bosonic (fermionic) field and Virasoro operators are given by a simple bilinear form of these fields. We shall show that the Virasoro condition correctly reproduces the Gromov-Witten invariants also in the case of manifolds with non-vanishing non-analytic classes ($h^{p,q}\not=0,p…
Virasoro constraints for K3 surfaces and monodromy operators
本研究利用Markman单值化算子证明K3曲面的Virasoro约束，将问题约化到秩1情形，同时证明秩0的新Virasoro约束，并引入负次数新型Virasoro算子，与已有算子共同构成中心荷为24的Virasoro代数表示。
For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\leq 1$ Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.
In this paper, we give a simple formula for the generating function of genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology, and use this formula to prove the genus-2 Virasoro conjecture for such manifolds.
A note on Virasoro constraints for products
研究当一个因子具有半简单量子cohomology时，积变分的Gromov-Witten理论的Virasoro约束
Virasoro constraints for topological recursion
本文针对任意谱曲线推导拓扑递归的祖先 Virasoro 约束，并在条件成立时建立后裔约束及非微扰推广，还通过实例比较其与几何不变量约束的关系。
M.Kazarian and S.Lando found a 1-parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP tau-functions. V.Bouchard and M.Marino suggested that this interpolation satisfies some deformed Virasoro constraints. However, they described the constraints in a somewhat sophisticated form of AMM-Eynard equations for the rather involved Lambert spectral curve. Here we present the relevant family of Virasoro constraints explicitly. They differ from the conventional continuous Virasoro constraints in the simplest possible way: by a constant s…
We will describe algebro-geometric solutions of the KdV hierarchy whose $τ$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten-Kontsevich $τ$-function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with ${\rm …
Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was discovered by Libgober and Wood that the product of the Chern classes $c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has been used by Eguchi, Jinzenji and Xiong in their approach to the quantum cohomology of $X$ via a representation of the Virasoro algebra with the central charge $c_n(X)$. In this paper we define for singular varieties $X$ a rational number $c_{st}^{1,n-1}(X)$ which is a stringy version of the number $c_1c_{n-1}$ for smooth $n$-folds. We show that the number $c_{st}^{1,n-1}(X)$ …
Virasoro Constraints for Orbifold Curves
本文证明了光滑射影有效轨形曲线相对Gromov-Witten理论的Virasoro约束，进而推导出其绝对Jiang-Tseng Virasoro猜想，为相关理论研究提供了关键支撑。
Virasoro constraints for Hodge integrals
该文研究任意目标簇Gromov-Witten理论中Hodge积分的Virasoro约束，提出相关猜想，证明零亏格约束、带单Hodge示性类插入的亏格1 $L_1^{\bbE}$ 约束，还得到高亏格Gromov-Witten不变量的新零化恒等式。
We first study the quantum product on the big phase space defined by gravitational Gromov-Witten invariants. We then use this product to give an interpretation for various topological recursion relations and also use it to study the Virasoro conjecture proposed by Eguchi-Hori-Xiong and Katz. We will give a recursive formulation to the Virasoro conjecture and study properties of relevent vector fields, which will be useful in proving and applying the Virasoro conjecture for all genera. In the genus-2 case, we will prove that the genus-2 Virasoro conjecture can be reduced to the $L_{1}$ constrai…
Virasoro constraints on moduli of sheaves and vertex algebras
本文将层理论 Virasoro 约束重新表述为 Joyce 顶点代数中的初态，证明其在壁穿越下保持不变，并借此证明了任意曲线上无挠层模空间及特定曲面上的 Virasoro 约束猜想。
We describe a geometric theory of Virasoro constraints in generalized Drinfeld-Sokolov hierarchies. Solutions of Drinfeld-Sokolov hierarchies are succinctly described by giving a principal bundle on a complex curve together with the data of a Higgs field near infinity. String solutions for these hierarchies are defined as points having a big stabilizer under a certain Lie algebra action. We characterize principal bundles coming from string solutions as those possessing connections compatible with the Higgs field near infinity. We show that tau-functions of string solutions satisfy second-order…
Cohomological Field Theory with vacuum and its Virasoro constraints
本文提出并部分验证了带真空的齐次上同调场论的Virasoro猜想，证明了半单情形的完整猜想，并应用于变形负r-自旋理论及dessins d'enfants理论的扩展。
Using new explicit formulas for the stationary GW/PT descendent correspondence for nonsingular projective toric 3-folds, we show that the correspondence intertwines the Virasoro constraints in Gromov-Witten theory for stable maps with the Virasoro constraints for stable pairs. Since the Virasoro constraints in Gromov-Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric 3-folds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.
Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections
该研究在射影空间光滑法诺完全交的ambient状态空间上证明了亏格一Virasoro猜想，推广了拟映射不变量的跨壁公式，为完全交与射影空间的Gromov-Witten理论建立了关键桥梁。
We introduce Virasoro operators for any Landau-Ginzburg pair (W, G) where W is a non-degenerate quasi-homogeneous polynomial and G is a certain group of diagonal symmetries. We propose a conjecture that the total ancestor potential of the FJRW theory of the pair (W,G) is annihilated by these Virasoro operators. We prove the conjecture in various cases, including: (1) invertible polynomials with the maximal group, (2) some two-variable polynomials with the minimal group, (3) certain Calabi-Yau polynomials with groups. We also discuss the connections among Virasoro constraints, mirror symmetry o…
In their fundamental work, B. Dubrovin and Y. Zhang, generalizing the Virasoro equations for the genus 0 Gromov-Witten invariants, proved the Virasoro equations for a descendent potential in genus 0 of an arbitrary conformal Frobenius manifold. More recently, a remarkable system of partial differential equations, called the open WDVV equations, appeared in the work of A. Horev and J. P. Solomon. This system controls the genus 0 open Gromov-Witten invariants. In our paper, for an arbitrary solution of the open WDVV equations, satisfying a certain homogeneity condition, we construct a descendent…
We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds CP^1 and CP^2 (or more generally, smooth projective curves and smooth simply-connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on the compactification of M_{g,n}. In particular, we show that the Virasoro conjecture for CP^2 implies the numerical part of Faber's conjecture on the tautological Chow ring of M_g.
We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter …
We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira (arXiv:2008.13746) and Moreira, Oblomkov, Okounkov and Pandharipande (arXiv:2008.12514). We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.
Virasoro constraints and representations for quiver moduli spaces
本文利用Joyce顶点代数研究带关系的箭图表示模空间的Virasoro约束，在光滑假设下构造了Virasoro代数的表示，并通过例外簇将其应用于del Pezzo曲面层模空间及Grassmannian。
We formulate Virasoro constraints for the generating functions of the intersection numbers on Hassett's moduli of weighted pointed curves and show that they are governed by the KdV integrable hierarchy.
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.
Virasoro constraints for orbifold Gromov-Witten theory are described. These constraints are applied to the degree zreo, genus zero orbifold Gromov-Witten potentials of the weighted projective stacks $\mathbb{P}(1,N)$, $\mathbb{P}(1,1,N)$ and $\mathbb{P}(1,1,1,N)$ to obtain formulas of descendant cyclic Hurwitz-Hodge integrals.
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…
We work through, in detail, the orbifold quantum cohomology, with gravitational descendants, of the stack BG, the point modulo trivial action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear in this theory which completely determine the correlators of the theory.
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…
Cohomology rings of the moduli of one-dimensional sheaves on the projective plane
本文研究射影平面上半稳定一维层模栈的上同调环，利用几何重言关系与 Virasoro 算子确定至 5 次的环结构，并验证局部 P² 的精细 GV/PT 对应及加强 P=C 猜想。
Universal Virasoro constraints for quivers with relations
本文提出箭图表示模空间 Virasoro 约束的通用方法，证明其对任意带关系、含冻结顶点且无环的有限箭图成立，并借助导出等价给出射影平面及双射影线上 Gieseker 半稳定层 Virasoro 约束的独立证明，进而独立证明任意曲面上秩 1 情形。
Bidifferentials, Lagrangian projections and the Virasoro extension
本文证明曲线 de Rham 上同调中 Lagrangian 补充与特定对称双微分一一对应，并确定其携带的上同调类，进而构造局部域导子李代数的 Virasoro 扩张。
Quantum Lefschetz theorem by Coates and Givental gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semi-simplicity and the Virasoro conjecture for the Gromov-Witten theories of (not necessarily Fano) projective toric manifolds.
Joyce's invariant and Virasoro Constraints for Quot schemes on curves
该研究为曲线上的商概型引入乔伊斯枚举不变量，结合顶点代数框架证明其Virasoro约束，并用于计算商概型上$f$-类的虚相交数。
This paper concerns the recent Virasoro conjecture for the theory of stable pairs on a 3-fold proposed by Oblomkov, Okounkov, Pandharipande and the author in arXiv:2008.12514. Here we extend the conjecture to 3-folds with non-$(p,p)$-cohomology and we prove it in two specializations. For the first specialization, we let $S$ be a simply-connected surface and consider the moduli space $P_n(S\times \mathbb{P}^1, n[\mathbb{P}^1])$, which happens to be isomorphic to the Hilbert scheme $S^{[n]}$ of $n$ points on $S$. The Virasoro constraints for stable pairs, in this case, can be formulated entirely…
Virasoro Constraints for Toric Bundles
本文证明 Gromov–Witten 理论中的 Virasoro 猜想对环面丛全空间成立当且仅当对底空间成立，通过局部化公式和非等变极限结合 Brown 镜像定理实现。
The Virasoro operations in Witten's theory of two-dimensional topological gravity have a homotopy-theoretic interpretation as endomorphisms of an ordinary cohomology theory with coefficients in a localization of I. Schur's ring Δof Q-functions. The resulting theory has many of the features of a vertex operator algebra.
This is a brief summary of our works [arXiv:1112.4063, arXiv:1201.4501] on constructing higher genus B-model from perturbative quantization of BCOV theory. We analyze Givental's symplectic loop space formalism in the context of B-model geometry on Calabi-Yau manifolds, and explain the Fock space construction via the renormalization techniques of gauge theory. We also give a physics interpretation of the Virasoro constraints as the symmetry of the classical BCOV action functional, and discuss the Virasoro constraints in the quantum theory.
A Gromov-Witten theory for simple normal-crossing pairs without log geometry
本研究定义了简单正规交叉除子的新型相对Gromov-Witten理论，证明其多项结构性质，并利用相对量子上同调零次部分给出Gross-Siebert替代镜像构造、证明Gross-Hacking-Keel的Frobenius结构猜想。
Deformations with cohomology constraints: a review
本文综述特征零域上由L∞对控制的带上同调约束的形变问题理论，并聚焦其近期应用。
We study constraints on the Chern classes of a vector bundle on a singular variety. We use this constraint to study a variety which carries a Hodge cycle that are not a linear combination of Chern classes of vector bundles on it.
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