论文标题
非定向表面角色品种的拓扑量子场理论
A Topological Quantum Field Theory for Character Varieties of Non-orientable Surfaces
论文作者
论文摘要
在本文中，我们研究了不可取向封闭表面的$ g $代表性和性格品种。通过基于拓扑量子场理论（TQFT）的几何方法，我们在Grothendieck of品种中计算了这些品种的虚拟类别，价格为$ g $，等于$ \ textrm {agl} _1 _1 $和$ \ \ \ textrm {sl} _2 $。在可定向的封闭表面的情况下，该方法已经知道和使用，我们将其扩展到不可取向表面的情况。此外，我们提供了一种实用的方法来显式计算TQFT，从而可以进行更简化和结构化的计算。最后，对于$ g = \ textrm {sl} _2 $，我们描述并解释了定向和不可定向的封闭表面的表示形式之间的关系。
In this paper, we study the $G$-representation and character varieties of non-orientable closed surfaces. By means of a geometric method based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of these varieties in the Grothendieck ring of varieties for $G$ equal to $\textrm{AGL}_1$ and $\textrm{SL}_2$. This method was already known and used in the case of orientable closed surfaces, and we extend it to the case of non-orientable surfaces. Furthermore, we provide a practical approach for explicitly computing the TQFT, allowing for more simplified and structured computations. Finally, for $G = \textrm{SL}_2$ we describe and explain the relationship between the representation varieties of the orientable and non-orientable closed surfaces.
