This is an old revision of this page, as edited by JarahE (talk | contribs) at 18:19, 18 July 2006 (+level, gauge group. CS only describes open strings in A model). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Revision as of 18:19, 18 July 2006 by JarahE (talk | contribs) (+level, gauge group. CS only describes open strings in A model)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)Chern-Simons theory is a topological gauge theory in three dimensions which describes knot and three-manifold invariants. It was introduced by Edward Witten in an attempt to interpret the Jones polynomial in terms of a three dimensional Yang-Mills theory. It is named so because its action is nothing but the Chern-Simons 3-form.
A particular Chern-Simons theory is specified by a choice of Lie group known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. The partition function of the quantum theory is only well-defined when the level is an integer.
This theory also makes the connection between these knots and 3-dimensional invariants and conformal field theory in 2 dimensions and in particular the Wess-Zumino-Witten model.
In the context of string theory, a U(N) Chern-Simons theory on an oriented Lagrangian 3-manifold M arises as the string field theory of open strings in the A-model topological string formulated on the cotangent bundle of M (The B-Model topological open string field theory being a the holomorphic Chern-Simons theory).
The information contained in Chern-Simons theory can also be captured through the use of matrix models.
This theory is gauge-invariant over boundary manifolds and manifolds with zero curvature form at their boundaries. However, this theory is gauge variant if there is no restriction on the curvature form at the boundary. This might be useful in anomaly inflow mechanisms.
This terms is nonlocal with respect to gauge invariant quantities.
The Chern-Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massive photon if this term is added to the Yang-Mills action. This term can be induced by integrating over a massive charged Dirac field. It also appears in the quantum Hall effect.
See also
References
- S.-S. Chern and J. Simons, "Characteristic forms and geometric invariants", Annals Math. 99, 48–69 (1974).
- Edward Witten, Quantum Field Theory and the Jones Polynomial, Commun.Math.Phys.121:351,1989.
- Edward Witten, Chern-Simons Theory as a String Theory, Prog.Math.133:637-678,1995.
- Marcos Marino, Chern-Simons Theory and Topological Strings, Rev.Mod.Phys.77:675-720,2005.
- Marcos Marino, Chern-Simons Theory, Matrix Models, And Topological Strings (International Series of Monographs on Physics), OUP, 2005.