Abstract

The study of representation varieties of fundamental groups of manifolds has long provided a beautiful bridge between algebraic geometry and different fields of Mathematics such as knot theory and mathematical physic. These kind of varieties have a very rich structure which, in many occasions, can be studied by relating it with the geometry of the manifold. This is specially obvious in the case of representation varieties of surfaces, whose Hodge structures have been observed to satisfy certain mirror-symmetric properties. But moreover, these Hodge structures can be directly related with the topology of the surface by using Topological Quantum Field Theories (TQFTs). In this talk we will recall how these TQFTs have been classically used to study representation varieties of surfaces and we will show how they can be used more generally for any manifold having an appropriate decomposition into bordisms, focusing on the case of complements of torus links in the three-dimensional sphere.