Topologization of electron liquids with Chern-Simons theory and quantum computation

Abstract

In 1987 a Geometry and Topology year was organized by Prof. Chern in Nankai and I participated as an undergraduate from the University of Science and Technology of China. There I learned about M. Freedman’s work on 4-dimensional manifolds. Then I went to the University of California at San Diego to study with M. Freedman in 1989, and later became his most frequent collaborator. It is a great pleasure to contribute an article to the memory of Prof. Chern based partially on some joint works with M. Freedman and others. Most of the materials are known to experts except some results about the classification of topological quantum field theories (TQFTs) in the end. This paper is written during a short time, so inaccuracies are unavoidable. Comments and questions are welcome. There are no better places for me to start than the Chern-Simon theory. In the hands of Witten, the Chern-Simons functional is used to define TQFTs which explain the evaluations of the Jones polynomial of links at certain roots of unity. It takes great imagination to relate the Chern-Simons theory to electrons in magnetic fields, and quantum computing. Nevertheless, such a nexus does exist and I will outline this picture. No attempt has been made regarding references and completeness.