Virtual Classes of Character Stacks

TL;DR

This work develops a stacky motivic framework to retain $G$-equivariant information when studying representation varieties by passing to character stacks $\,\mathfrak X_G(M)=[R_G(M)/G]$ and their motives in the Grothendieck ring of representable stacks over ${\textup{B}G}$. It constructs a symmetric lax monoidal TQFT $Z=\mathcal Q\circ\mathcal F$ from bordisms to $\textup{K}(\textup{\bf RStck}/{\textup{B}G})$-modules, enabling effective computation of the virtual classes $[\mathfrak X_G(W)]$ for closed orientable manifolds, with detailed two-dimensional genus-$g$ applications. The paper gives explicit motivic formulas for character stacks when $G=\textup{AGL}_1(k)$ and for $G=\mathbb{G}_m \rtimes \mathbb{Z}/2\mathbb{Z}$, including how to extract representation-variety data via the evaluation map to $\hat{\textup{K}}(\textup{\bf Var}_k)$. It further highlights phenomena for non-connected groups, where naive point-counting fails and the equivariant motive carries richer information, demonstrated by concrete computations and Luna-type stratifications. Overall, the stacky TQFT provides a robust toolkit for equivariant motivic invariants of representation- and character-theoretic moduli.

Abstract

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.

Theorems & Definitions (71)

• Theorem
• Theorem
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7