On a Proof of a Conjecture of Marino-Vafa on Hodge Integrals

The Laplace transform of the cut-and-join equation of Mariño-Vafa formula and its applications

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…

Formulae of one-partition and two-partition Hodge integrals

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…

Hodge integrals, Hurwitz numbers, and Symmetric Groups

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.

Hodge integrals and Gromov-Witten theory

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…

Hodge integrals and degenerate contributions

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.

A Conjecture on Hodge Integrals

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.

Local Mirror Symmetry for the Topological Vertex

For three-partition triple Hodge integrals related to the topological vertex, we derive Eynard-Orantin type recursion relations from the cut-and-join equation. This establishes a version of local mirror symmetry for the local $C^3$ geometry with three D-branes, as proposed by Marino and Bouchard-Klemm-Marino-Pasquetti.

Hodge integrals and invariants of the unknot

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.

On blowup formulae for the S-duality conjecture of Vafa and Witten II: the universal functions

This is the second part of two papers. In this part, we establish closed formulae for the universal functions in the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck compactification spaces over algebraic surfaces.

从框架sheaves的墙穿越考虑Vafa-Witten不变量

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。

On blowup formulae for the S-duality conjecture of Vafa and Witten

This is the first part of two papers. In this part, we prove the blowup formulae for virtual Hodge polynomials of Gieseker moduli spaces of rank-2 stable sheaves and Uhlenbeck compactification spaces over algebraic surfaces. In particular, we verify the blowup formulae for the S-duality conjecture of Vafa-Witten, i.e. the blowup formulae for the Euler numbers of instanton moduli spaces over algebraic surfaces.

Hodge-GUE correspondence and the discrete KdV equation

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.

Localization and Conjectures from String Duality

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…

希格斯丛与Gopakumar-Vafa不变量的上同调χ-独立性

Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants

本文证明了局部曲线Gopakumar-Vafa不变量的Toda χ-独立性猜想，并引入不依赖χ的希格斯丛模空间BPS上同调，将Hausel-Thaddeus猜想推广到混合Hodge模同构层面且无需互素假设。

Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces

Let (S,H) be a rational algebraic surface with an ample divisor. We compute generating functions for the Hodge numbers of the moduli spaces of H-stable rank 2 sheaves on S in terms of certain theta functions for indefinite lattices that were introduced in the paper alg-geom/9612020 written jointly with Don Zagier. If H lies in the closure of the ample cone and has self-intersection 0, it follows that the generating functions are Jacobi forms. In particular the generating functions for the Euler numbers have a similar transformation behaviour under SL(2,Z) as that predicted in Vafa and Witten: …

ADKMV猜想的一个证明——拓扑顶点的3-KP可积性

On a Proof of the ADKMV Conjecture -- $3$-KP Integrability of the Topological Vertex

本文通过引入推广的玻色-费米场赋值，证明了拓扑顶点生成函数满足3分量KP层级的ADKMV猜想，并将其推广到带框架拓扑顶点及所有光滑环面Calabi-Yau三维流形。

Conjectures on spectral numbers for upper triangular matrices and for singularities

Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral numbers $α_1,...,α_n\in{\mathbb R}$ to real upper triangular $n\times n$ matrices $S$ with 1's on the diagonal and eigenvalues of $S^{-1}S^t$ in the unit sphere. Especially, $\exp(-2πiα_j)$ shall be the eigenvalues of $S^{-1}S^t$. We tried to make their idea rigorous, but we succeeded only partially. This paper fixes our results and our conjectures. For certain subfamilies of matrices their idea works marvellously, and there the spectral numbers fit well to natural (split) polarized mixed Hodge structures. We formulat…

On the motivic stable pairs invariants of K3 surfaces

For a K3 surface S, we study motivic invariants of stable pairs moduli spaces associated to 3-fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of Kawai-Yoshioka, imply a full determination of the theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The work may be viewed as the third in a sequence of formulas starting with Yau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical data suggest the motivic invariants are linked to the Mathieu M_…