Field theory of incompressible topological quantum liquids in higher dimensions

I will discuss a derivation of field theories that describe quantum dynamics of topological defects in a broad range of topologically ordered phases. Beyond quantum Hall liquids in two dimensions, the obtained Lagrangians hint at the existence of stable topologically ordered states with fractional excitations in higher dimensions - involving defects such as monopoles and hedgehogs. I will outline some properties of these topological phases (e.g. fractionalized magnetoelectric effect), and touch upon the prospects for their observation in three-dimensional topological materials.