Applications of Chern-Simons Theory

In the early 1970s mathematicians S.S. Chern and Jim Simons collaborated on a paper, “Characteristic Forms and Geometric Invariants,” inspired by a failed effort by Jim to find a combinatorial formula for the signature of a 4-manifold. Out of that work a term related to 3-manifolds emerged with some very interesting properties, and when Chern saw this he realized it could be vastly generalized – thus the paper referenced above. That paper led to additional interesting mathematics such as differential cohomology, but it never occurred to its authors that so called Chern-Simons theory would subsequently play a significant role in a number of areas of physics.

On April 25, Simons briefly described his work with Chern, followed by three talks on different areas of physics that involve Chern-Simons theory.

Cumrun Vafa, the Donner Professor of Science at Harvard University, will talk about the impact of Chern-Simons theory on topological aspects of string theory. He will review how the theory has led, among other things, to profound advances in enumerative geometry.

Michael Freedman will discuss Jim’s topological term and quantum computing. Freedman is director of Station Q, Microsoft’s project on quantum physics and quantum computation located on the campus of the University of California, Santa Barbara. He will discuss how the Chern-Simons theory opens a direct path to topological physics and can help bring about quantum computing. Experiments are underway to show that topological physics derived from SU(2)_2 Chern-Simons theory will supply the stability missing from conventional qubit systems. The SU(2)_2 theory is closely related to the Ising theory and Majorana fermions, Jones polynomial of links at 4-th root of unity, and the fractional quantum Hall state at filling fraction 5/2.

Charles Kane, the Christopher H. Browne Distinguished Professor of Physics at the University of Pennsylvania, will present on Chern-Simons theory in condensed matter physics from topological band theory to topological field theory. He will demonstrate how the mathematical constructions introduced by Chern and Simons have had a profound impact on our understanding of quantum phases of matter, helping to inspire the discoveries of new classes of electronic materials that have significant implications for both fundamental science and technology.