Representation variety for the rank one affine group

Abstract

The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group.

arXiv comment28 pages, 3 figures. References added

References in corpus (10)

• Mirror Symmetry is T-Duality
• Mirror symmetry, Langlands duality, and the Hitchin system
• Singularities of free group character varieties
• On the E-polynomials of a family of Character Varieties
• Arithmetic of singular character varieties and their $E$-polynomials
• Virtual classes of parabolic ${\mathrm{SL}}_2(\mathbb{C})$-character varieties
• A lax monoidal Topological Quantum Field Theory for representation varieties
• Varieties of Characters
• Motivic theory of representation varieties via Topological Quantum Field Theories
• E-polynomials of $PGL(2,\mathbb{C})$-character varieties of surface groups

Cited by in corpus (2)

• Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories
• Pseudo-quotients of algebraic actions and their application to character varieties