论文标题
字符堆栈的虚拟类别
Virtual Classes of Character Stacks
论文作者
论文摘要
在本文中，我们扩展了由冈萨雷斯 - 普里埃托（González-Prieto），Gogares和Muñoz开发的拓扑量子场理论，用于计算$ g $ - 代表品种的虚拟类别的封闭式表面，可在特色环中的封闭方向表面，以供角色堆栈的设置。为此，我们定义了一个合适的代表堆栈的Grothendieck环，在该堆栈中定义了拓扑量子场理论。通过这种方式，我们计算了字符堆叠的虚拟类别，即$ bg $，也就是说，在自然伴随动作方面，代表性变种的动机分解。 我们在两种情况下应用了此框架，为封闭的任意属的封闭方向表面的字符堆栈的虚拟类别提供了明确的表达式。首先，在级别$ 1 $的仿射线性群体中，字符堆栈的虚拟类别完全记住了自然的伴随动作，尤其可以直接得出字符变体的虚拟类别。其次，我们考虑了非连接的$ \ mathbb {g} _m \ rtimes \ mathbb {z}/2 \ mathbb {z} $，我们的理论如何允许我们计算角色堆栈的动机信息，而经典的naïve点点数计数方法失败了。
In this paper, we extend the Topological Quantum Field Theory developed by González-Prieto, Logares, and Muñoz for computing virtual classes of $G$-representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks. To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field Theory is defined. In this way, we compute the virtual class of the character stack over $BG$, that is, a motivic decomposition of the representation variety with respect to the natural adjoint action. We apply this framework in two cases providing explicit expressions for the virtual classes of the character stacks of closed orientable surfaces of arbitrary genus. First, in the case of the affine linear group of rank $1$, the virtual class of the character stack fully remembers the natural adjoint action, in particular, the virtual class of the character variety can be straightforwardly derived. Second, we consider the non-connected group $\mathbb{G}_m \rtimes \mathbb{Z}/2\mathbb{Z}$, and we show how our theory allows us to compute motivic information of the character stacks where the classical naïve point-counting method fails.
