Jian Zhou
We introduce a deformed topological vertex and use it to define deformations of the topological string partition functions of some local Calabi-Yau geometries. We also work out some examples for which such deformations satisfy a deformed Gopakumar-Vafa integrality and can be identified with the equivariant indices of some naturally defined bundles on the framed moduli spaces.
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$q$-Deformed Topological Recursion, Weight Vectors and Algebraic Structures
本文通过研究$q$-变形的$\mathcal{W}$-代数表示中的最高权重向量，推导出$q$-变形的拓扑递归，并证明其产生$q$-变形的量子曲线，统一了量子可积性的多种方法。
Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts
本文基于临界CoHAs构造顶点coproduct，证明其形成顶点双代数，并通过Majid-Radford bosonisation扩展协代数以包含Cartan部分，在ADE图中恢复Drinfeld变形coproduct，同时提供对称图协整性定理的新证明。
The Gromov-Witten invariants of \mathbb{C}^3 with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al in the fermionic picture. We will propose a similar formula for the framed topological vertex and prove it in the case when there are one or two branes.
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.
We prove the Bouchard-Mariño Conjecture for the framed one-legged topological vertex by deriving the Eynard-Orantin type recursion relations from the cut-and-join equation satisfied by the relevant triple Hodge integrals. This establishes a version of local mirror symmetry for the local $C^3$ geometry with one $D$-brane.
Gromov-Witten invariants and membrane indices of fivefolds via the topological vertex
通过拓扑顶点方法，研究五重叠空间的Gromov-Witten理论和膜指数，证明局部反对角拓扑动作下的猜想。
On one-leg orbifold topological vertex in refined Gromov-Witten theory
本文研究了精炼格罗莫夫-威滕理论中单腿轨道叠顶点的定义与计算，验证了其在光滑情况下的结果，并应用于局部足球的不变量计算。
$q$-Deformed Topological Recursion: Quantum Curves and Non-perturbative Analysis
该研究探究$(r,s)$-Airy结构的q-形变，构造q-WKB解，分类容许$(r,s,q)$对及q-卡西米尔构型，建立量子谱曲线与可积系统的关联，为镜像曲线q-量子化提供新见解。
We will establish the fermionic gluing principle of the topological vertex, that is, provided the framed ADKMV conjecture, the generating functions of the Gromov-Witten invariants of all toric Calabi-Yau threefolds are Bogoliubov transforms of the vacuum.
We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.
A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of…
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。
The topological string partition function for the neighbourhood of three spheres meeting at one point in a Calabi-Yau threefold, the so-called 'closed topological vertex', is shown to be reproduced by a simple Calabi-Yau crystal model which counts plane partitions inside a cube of finite size. The model is derived from the topological vertex formalism. This derivation can be understood as 'moving off the strip' in the terminology of hep-th/0410174, and offers a possibility to simplify topological vertex techniques to a broader class of Calabi-Yau geometries. To support this claim a flop transi…
We prove transformation formulae for generating functions of Gromov-Witten invariants on general toric Calabi-Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric Calabi-Yau threefolds.
We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of CP^3 and then to compute those invariants via the geometry of the Cremona transformation.
We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed…
The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and repres…
We compute the motivic Donaldson-Thomas theory of the resolved conifold, in all chambers of the space of stability conditions of the corresponding quiver. The answer is a product formula whose terms depend on the position of the stability vector, generalizing known results for the corresponding numerical invariants. Our formulae imply in particular a motivic form of the DT/PT correspondence for the resolved conifold. The answer for the motivic PT series is in full agreement with the prediction of the refined topological vertex formalism.
We compute the Donaldson-Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in arXiv:1603.05271, we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz-Klemm-Va…
We show that trace functions on modules of topological N=2 super vertex algebras give rise to conformal blocks on elliptic supercurves. We show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves. Under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector bundle of conformal blocks. We show that this connection is equivariant with respect to a natural action of the Jacobi modular group on the modular parameters and the trace functions. In the append…
We define Donaldson-Thomas invariants of Calabi-Yau orbifolds and we develop a topological vertex formalism for computing them. The basic combinatorial object is the orbifold vertex, a generating function for the number of 3D partitions asymptotic to three given 2D partitions and colored by representations of a finite Abelian group G acting on C^3. In the case where G=Z_n acting on C^3 with transverse A_{n-1} quotient singularities, we give an explicit formula for the vertex in terms of Schur functions. We discuss applications of our formalism to the Donaldson-Thomas Crepant Resolution Conject…
We present an orbifold topological vertex formalism for PT invariants of toric Calabi-Yau 3-orbifolds with transverse $A_{n-1}$ singularities. We give a proof of the orbifold DT/PT Calabi-Yau topological vertex correspondence. As an application, we derive an explicit formula for the PT $\mathbb{Z}_{n}$-vertex in terms of loop Schur functions and prove the multi-regular orbifold DT/PT correspondence.
We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorifying quantum group invariants a…
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