Electron fractionalization and cuprate superconductivity

We discuss the possibility that the electron may be fractionalized in some quantum phases of matter in two or higher dimensions. We review the theory of such phases, and show that their effective theory is a $Z_2$ gauge theory. These phases may be characterized theoretically through the notion of topological order. We discuss the appeal of fractionalization ideas for building theories of the cuprates, and possible ways of testing these ideas.