Knotty character varieties

We study constructible sheaves on a topological surface with singular support prescribed by a legendrian knot in the unit cotangent bundle. As we explain, moduli spaces of such include the wild character varieties which parameterize connections with irregular singularities, but also many more spaces. We explore connections to known constructions in legendrian knot theory, to Khovanov-Rozansky knot homology, and to the P=W conjecture in nonabelian hodge theory.