Topological Quantum Field Theories for character varieties

Abstract

This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro-geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we develop a formalism that allows us to construct TQFTs from two simpler pieces of data: a field theory (geometric data) and a quantisation (algebraic data). As an application, we construct a TQFT computing Hodge structures on representation varieties and we use it for computing explicity the Deligne-Hodge polynomials of parabolic ${\mathrm{SL}}_2(\mathbb{C})$-character varieties.

arXiv comment202 pages, 18 figures, PhD thesis (Madrid, 2018)

References in corpus (1)

• The Symmetric Tensor Product on the Drinfeld Centre of a Symmetric Fusion Category

Cited by in corpus (4)

• Virtual classes of parabolic ${\mathrm{SL}}_2(\mathbb{C})$-character varieties
• Motivic theory of representation varieties via Topological Quantum Field Theories
• 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