论文标题
$ \ mathbb c $ - 线性枚举不变理论中的通用结构
Universal structures in $\mathbb C$-linear enumerative invariant theories
论文作者
论文摘要
代数几何形状，差异几何或代表理论中的列举不变理论是对不变的研究的研究，该研究“计数” $τ$ - （半）稳定的对象$ e $与固定拓扑不变的$ [e] =α$在某些几何学上，使用虚拟类别$ [ SS}（τ）] _ {\ rm virt} $在某些同源性理论中，用于模量空间$ {\ cal m}_α^{\ rm st}（\ rm st}（τ）\ subseteq {\ cal m}_α^_α^{\ rm ss}（\ rm ss}（\ rm ss}（τ）$的$ - 例子包括莫希济族的不变式计数在表面上的连贯的滑轮，Donaldson-Thomas型不变式在Calabi-yau上计数3倍和4倍的连贯的滑轮，而Fano 3倍，以及4个Manifolds的Donaldson Invariants。 我们对许多列举不变理论共有的新通用结构做出了猜想。这样的理论有两个模块空间$ {\ cal m}，{\ cal m}^{\ rm pl} $，其中第二作者给出$ h _**（{\ cal m}）$逐步的顶点代数的结构$ h _*（{\ cal m}）$。虚拟类$ [{\ cal m}_α^{\ rm ss}（τ）] _ {\ rm virt} $在$ h _**（{\ cal m}^{\ cal m}^{\ rm pl}）中取值。 定义$ [{\ cal m}_α^{\ rm ss}（τ）]还原）是一个困难的问题。我们指出，有一种自然的方法可以定义$ [{\ cal m}_α^{\ rm ss}（\ rm ss}（τ）] _ {\ rm virt} $在$ \ mathbb q $上的同源性中m}^{\ rm pl}）$。我们证明了我们对无定向循环的震颤表示的模量空间的猜想。 我们使用Behrend-Fantechi虚拟类中的代数几何形状的猜想在续集ARXIV中证明了：2111.04694。
An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $τ$-(semi)stable objects $E$ with fixed topological invariants $[E]=α$ in some geometric problem, using a virtual class $[{\cal M}_α^{\rm ss}(τ)]_{\rm virt}$ in some homology theory for the moduli spaces ${\cal M}_α^{\rm st}(τ)\subseteq{\cal M}_α^{\rm ss}(τ)$ of $τ$-(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Such theories have two moduli spaces ${\cal M},{\cal M}^{\rm pl}$, where the second author gives $H_*({\cal M})$ the structure of a graded vertex algebra, and $H_*({\cal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\cal M})$. The virtual classes $[{\cal M}_α^{\rm ss}(τ)]_{\rm virt}$ take values in $H_*({\cal M}^{\rm pl})$. Defining $[{\cal M}_α^{\rm ss}(τ)]_{\rm virt}$ when ${\cal M}_α^{\rm st}(τ)\ne{\cal M}_α^{\rm ss}(τ)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\cal M}_α^{\rm ss}(τ)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $τ$, written using the Lie bracket on $H_*({\cal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Our conjectures in Algebraic Geometry using Behrend-Fantechi virtual classes are proved in the sequel arXiv:2111.04694.
