Magdalena Boos, Hans Franzen
We study torus actions on moduli spaces of quivers. First we give a description of the weight spaces of the induced action of the tangent space to a torus-fixed point. Then we focus on actions of tori of rank one and derive an explicit form for the attractors in the Białynicki-Birula decomposition.
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We compute rational equivariant Chow rings with respect to a torus of quiver moduli spaces. We derive a presentation in terms of generators and relations, use torus localization to identify it as a subring of the Chow ring of the fixed point locus, and we compare the two descriptions.
Desingularizations of Quiver Grassmannians for the Equioriented Cycle Quiver
该研究针对等向循环箭图的幂零表示，构造了箭图格拉斯曼流形的环面等变去奇异化方法，并将其用于计算相应的环面等变上同调。
In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.
Skeletal Torus Actions and GKM Structures on Quiver Grassmannians of String Representations
本文研究等定向A型与幂零等定向Ã型箭图表示的箭图格拉斯曼流形，证明其为GKM簇，并对弦表示情形下具有GKM簇结构的所有相应箭图格拉斯曼流形给出完整分类。
Torus fixed points of quiver moduli spaces are given by stable representations of the universal (abelian) covering quiver. As far as the Kronecker quiver is concerned they can be described by stable representations of certain bipartite quivers coming along with a stable colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. For certain roots it is even possible to construct all torus fixed points.
The fixed point set under a natural torus action on projectivized moduli spaces of simple representations of quivers is described. As an application, the Euler characteristic of these moduli is computed.
Semisimple representations of the free product Z_p*Z_q determine θ-semistable representations of a specific quiver Q_pq. The dimension vectors of θ-stable representations of this quiver were classified by Le Bruyn and Adriaenssens. In this paper we classify the moduli spaces M_α^ss(Q_pq,θ) which are smooth projective varieties.
Białynicki-Birula Decompositions of Nakajima Quiver Varieties, Quiver Chains and Star-Shaped Quivers
研究中岛箭图簇相关的比亚林斯基 - 比鲁拉分解，通过箭图链描述不动点，探讨分解几何，如吸引纤维维度等，还将框架用于星形箭图，对特定维数向量分类不动点轨迹分量并计算动机类。
Moduli spaces of representations of quivers with multiplicities via non-reductive GIT
该研究利用非约化群的相对仿射GIT与新稳定性条件，构造了截断幂级数环上带重数的箭图表示模空间（含带标架模空间及Nakajima箭图簇类似物），并证明部分光滑模空间上同调纯净。
Gromov-Witten invariants of weighted projective planes and Euler characteristics of moduli spaces of representations of bipartite quivers are related via the tropical vertex, a group of formal automorphisms of a torus. On the Gromov-Witten side, this uses the work of Gross, Pandharipande and Siebert. The quiver moduli side features quiver wall-crossing formulas, functional equations for Euler characteristics, and localization techniques. We derive several explicit formulas for Gromov-Witten invariants.
Euler Characteristics of Generic Quiver Grassmannians: Semi-Invariants and Localization
研究有限无圈箭图\(Q\)的一般箭图格拉斯曼流形\(\operatorname{Gr}_\beta(V)\)，通过结合多种公式将其拓扑欧拉特征表示为协变重数的有限带符号和，给出任意\(Q\)的有限环面局部化公式，并在广义克罗内克箭图情形下进行说明。
By a result of Klyachko the Euler characteristic of moduli spaces of stable bundles of rank two on the projective plane is determined. Using similar methods we extend this result to bundles of rank three. The fixed point components correspond to moduli spaces of the subspace quiver. Moreover, the stability condition is given by a certain system of linear inequalities so that the generating function of the Euler characteristic can be determined explicitly.
Let $X$ be a Nakajima quiver variety and $X'$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes \mathbb{C}^{\times}$ on $X'$. Assuming that $(X')^{\mathsf{K}}$ is finite, we propose a formula for the $\mathsf{K}$-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of $X$ known as vertex functions.
Nakajima quiver bundles
本文提出Nakajima束表示概念，结合代数曲线与黎曼面，推广了普通Nakajima quiver表示与quiver bundles，通过 gauge理论构造 generalized quiver 变分，并研究其变形理论与稳定性对应关系。
In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Groebner bases and toric geometry. For a finite abelian group G in GL(n,k), let Y_θbe the coherent component of the moduli space of θ-stable representations of the McKay quiver. Our two main results are as follows: we provide a simple description of the quiver representations corresponding to the torus orbits of Y_θ, and, in the case where Y_θequals Nakamura's G-Hilbert scheme, we present explicit equations for a cover by local coordinate charts. The latter …
Flags of sheaves, quivers and symmetric polynomials
本文研究了嵌套瞬子箭图的表示理论，证明了其稳定表示的模空间为嵌套Hilbert概形和层旗提供了类似ADHM的构造，并利用等变局部化计算了拓扑不变量，猜想其全纯欧拉示性数的生成函数与Macdonald多项式相关。
The 4-fold Pandharipande--Thomas vertex
本文提出环面Calabi--Yau 4重上Pandharipande--Thomas稳定对等变顶点的完整显式猜想描述，利用箭图格拉斯曼流形识别不动点轨迹，并规定切-障碍理论的一半，支持低次数计算机验证DT/PT对应。
The geometry of Nekrasov's gauge origami theory
本文提出通过代数几何方法定义折纸配分函数，并探讨其在非微扰Dyson-Schwinger方程中的应用
Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ …
We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for t…
Modelling $A$-branes with foliations
本文通过叶状结构框架描述环面Calabi-Yau三维流形镜流形中的$A$-膜，基于胞腔分解显式构建其模空间，并揭示特殊拉格朗日流形手术参数与$m$-Kronecker箭图表示论间的局部映射关系。
Bow varieties as symplectic reductions of $T^*(G/P)$
本文将每个具有环面不动点的弓形簇实现为部分旗簇余切丛的幂幺群辛约化，并利用该描述计算了部分弓形簇的等变上同调，回答了先前问题，同时在附录中给出了显式广义 Mirkovic-Vybornov 同构。
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