Virtual classes of character stacks

• DOI: 10.1016/j.geomphys.2025.105450
• Publication Date: 2025-02-14T21:09:00Z
• Authors: Ángel González-Pr...; Márton Hablicsek; Jesse Vogel
• Keywords: Mathematics - Algebraic Geometry; FOS: Mathematics; 57R56, 14D23, 14M35; 0101 mathematics; Representation Theory (math.RT); 01 natural sciences; Algebraic Geometry (math.AG); Mathematics - Representation Theory

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.<br/>38 pages. Comments are welcome!<br/>