Q-Math Seminar
Topological Quantum Field Theories for Character Groupoids
Speaker: Ángel González-Prieto (UCM)Date: Tuesday, 06 May 2025 - 11:00Place: Room 2.2.D08, Universidad Carlos III de Madrid
Online: https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380 (active on request)
Abstract:
Given a compact manifold M, the space of representations of its fundamental group into a group G plays a central role in modern geometry, reflecting deep connections between the topology of M and the representation theory of G. When G is finite, this space naturally acquires the structure of a groupoid—known as the character groupoid—whose combinatorics are rich and intricate.
In this talk, we construct a Topological Quantum Field Theory (TQFT) that computes the number of points of these character groupoids, extending the celebrated Dijkgraaf–Witten theory. As we shall see, this TQFT provides a complete encoding of the classical representation theory of finite groups. We will also address the case where G is an algebraic group, for which the space of representations becomes the so-called character stack. We will explain how the TQFT developed in the finite case extends naturally to this algebraic setting, leading to a far-reaching generalization of character theory in terms of algebraic K-theory and character sheaves.
