通过稳定对象的曲线计数理论I。DT/PT通信

N=2经典规范群规范理论的瞬子计数

A note on instanton counting for N=2 gauge theories with classical gauge groups

利用Nekrasov提出的瞬子计数方法研究了N=2规范理论的前势。对于无物质的SO理论，我们得到了全预势及其弦论引力修正的封闭表达式。对于更微妙的情况下，SP理论没有问题，我们讨论一般的功能和计算的prepotential到瞬子数3。我们也简要地讨论了SU理论与物质的对称和反对称表示。我们检查我们所有的结果对相应的Seiberg-Witten几何的预测。

The DT/PT correspondence for smooth curves

We show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi-Yau threefold. We exploit a local study of the Hilbert-Chow morphism about the cycle of a smooth curve. We determine, via Quot schemes, the global Donaldson-Thomas theory of a general Abel-Jacobi curve of genus $3$.

ADE McKay箭图的计数不变量

Counting invariants for the ADE McKay quivers

本文研究了二元多面体群G < SU（2）< SU（3）的McKay表示的模空间.这种表示的导出范畴等价于在相应的ADE分解Y=G-Hilb（C^3）上的相干层的导出范畴。根据Nagao和Nakajima的思想，通过对等价的导出范畴的稳定性条件空间中的参数进行特殊的选择，我们恢复了Donaldson-Thomas（DT），Pandharipande-Thomas（PT）和Szendroi（NCDT）模空间.我们还直接计算了Y的Gromov-Witten（GW）分拆函数，并将结果用相应的ADE Dynkin图的根系表示.然后，我们通过假设DT/PT对应关系来验证GW/DT/NCDT对应关系。Szendroi不变量与Bryan定义的C^3/G的轨道Donaldson-Thomas不变量相同。这将使我们能够验证在这种情况下的轨道Donaldson-Thomas理论的Crepant归结猜想。

Counting (tropical) curves via scattering.sage

In this note I will explain how relative/log Gromov-Witten invariants of pairs $(X,D)$ with very ample smooth anticanonical divisor $D$ can be computed using algebro-combinatorial objects called scattering diagrams. The underlying principle behind this computational method is a tropical correspondence theorem for non-toric cases, which I will explain briefly. By computing some examples I will give an introduction to a sage code that I wrote for computing scattering diagrams.

Elliptic genus of phases of N=2 theories

We discuss an algebro-geometric description of Witten's phases of N=2 theories and propose a definition of their elliptic genus provided some conditions on singularities of the phases are met. For Landau-Ginzburg phase one recovers elliptic genus of LG models proposed in physics literature in early 90s. For certain transitions between phases we derive invariance of elliptic genus from an equivariant form of McKay correspondence for elliptic genus. As special cases one obtains Landau-Giznburg/Calabi-Yau correspondence for elliptic genus of weighted homogeneous potentials as well as certain hybr…

规范理论中BPS算子的计数：箭图、合冲图和体积图

Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics

我们发展了一个系统的和有效的方法计数单迹和多迹BPS运营商与两个超荷，世界体积规范理论的$N$ D膜探测器的$N \to \infty$和有限的$N$。该技术适用于一般奇点，orbifold，复曲面，非复曲面，完全相交，等等，甚至几何的精确场论的解释还不知道。所谓的“Plethystic Exponential”在（1）Calabi-Yau的定义方程，（2）单迹BPS算子的生成函数和（3）多迹算子的生成函数之间提供了一个简单的桥梁。在数学上，规范理论、代数几何、组合数学和数论之间迷人而复杂的相互关系以体积学和合朔学的形式展现出来。

局部空间上的拓扑规范理论与黑洞熵计数

Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings

研究复流形上全纯线丛全空间上的上同调规范理论，利用路径积分的U（1）等变局部化，得到了它们在基流形上的约化.我们通过精确的路径积分局部化证明了这种一般机制，减少了在hep-th/0411280中描述的局部曲线，与黑洞熵/Gromov-Witten不变量的计算有关。与四维规范理论的协议是恢复考虑在后者的非平凡的贡献，来自单圈波动决定因素的边界处的总空间。我们还研究了一类卡-丘局部曲面上的阿贝尔规范理论，描述了A模型的量子泡沫，与Donaldson-Thomas不变量的计算有关。

Note on the motivic DT/PT correspondence and the motivic Flop formula

We prove the motivic version of the DT/PT-correspondence in \cite{PT} and the motivic flop formula of the curve counting invariants in the derived category of smooth Calabi-Yau threefold DM stacks. The main method we use is Bridgeland's Hall algebra identities and the motivic integration map of Bridgeland and Joyce from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold DM stack $Y$ to the motivic quantum torus, which is a Poisson algebra homomorphism.

一种新的非阿贝尔规范理论的McKay对应

A new kind of McKay correspondence from non-Abelian gauge theories

Kähler流形$X$上二维规范线性西格玛模型的边界手征环对拓扑D膜扇区和它们之间的无质量开弦进行了分类.虽然它是在小体积下由简单群论确定的，但它对一般体积的延拓提供了关于$X$上的$D$ -膜的高度非平凡的信息，与导出的范畴$D^\flat(X)$有关。我们用这个对应详细说明了一个扩展的概念麦凯对应，捕捉更一般的比orbifold奇点。作为例子，我们对一类与格拉斯曼算子有关的非紧Calabi-Yau奇点给出了McKay对应的新概念。

轨道振子的瞬子计数和穿壁

Instanton counting and wall-crossing for orbifold quivers

阿贝尔轨道折叠奇点的非对易Donaldson-Thomas不变量可以通过在六维非对易N=2规范理论中计数瞬子解来研究;这种构造基于广义McKay对应，并将瞬子计数与自然地与奇点几何相关联的子的框架表示的计数等同起来。我们扩展这些建设计算BPS配分函数的高阶精炼和motivic非交换Donaldson-Thomas不变量的库仑分支的规范理论变量和orbifold数据。我们引入虚瞬子的概念，与自然的辛电荷晶格，其中管理的量子跨壁行为的BPS状态在这种情况下。McKay对应自然地将我们的形式主义与基于量子单值算子和簇代数的其他穿墙方法联系起来。

超对称规范理论的库仑分支的数学定义及Kac-Moody李代数的几何Satake对应关系

A mathematical definition of Coulomb branches of supersymmetric gauge theories and geometric Satake correspondences for Kac-Moody Lie algebras

本文提出3d N=4超对称规范理论库仑分支的数学定义，并基于库仑分支建立Kac-Moody李代数的几何Satake对应关系。

Counting tropical rational space curves with cross-ratio constraints

This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio condit…

弯曲空间上的Bethe/Gauge对应

Bethe/Gauge correspondence on curved spaces

Bethe/gauge对应确定了在二维N = 2 Poincare超对称下不变量的质量规范理论的超对称真空与某些量子可积系统的定态。超对称理论可以用多种方式扭曲，产生拓扑场论。对于这些理论，我们计算柄胶合算子H。我们还讨论了关于Bethe态范数的Gaudin猜想及其与H.

Correspondence Theorem between Holomorphic Discs and Tropical Discs on K3 Surfaces

We prove that the open Gromov-Witten invariants on K3 surfaces satisfy the Kontsevich-Soibelman wall-crossing formula. One one hand, this gives a geometric interpretation of the slab functions in Gross-Siebert program. On the other hands, the open Gromov-Witten invariants coincide with the weighted counting of tropical discs. This is an analog of the corresponding theorem on toric varieties \cite{M2}\cite{NS} but on compact Calabi-Yau surfaces.

The combinatorial PT-DT correspondence

We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions. The main tools in our proof are a Desnanot-Jacobi-type "condensation" identity, and a novel application of the tripartite double-dimer model of Kenyon-Wilson.