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| Chern-Simons theory is a |
Chern-Simons theory is a 3-dimensional topological quantum field theory of ] which computes ] and ] invariants. It was introduced by ] in an attempt to interpret the ] in terms of a three dimensional ]. It is named so because its ] is nothing but the ]. | ||
| A particular Chern-Simons theory is specified by a choice of ] known as the gauge group of the theory and also a number referred to as the <i>level</i> of the theory, which is a constant that multiplies the ]. The ] of the ] theory is |
A particular Chern-Simons theory is specified by a choice of ] G known as the gauge group of the theory and also a number referred to as the <i>level</i> of the theory, which is a constant that multiplies the ]. The action is gauge dependent, however the ] of the ] theory is ] when the level is an integer and the gauge ] vanishes on all ] of the 3-dimensional spacetime. | ||
| ==Relations with other theories== | |||
| This theory also makes the connection between these knots and 3-dimensional invariants and ] in 2 dimensions and in particular the ]. | |||
| ===Topological string theories=== | |||
| ⚫ | In the context of ], a U(N) Chern-Simons theory on an oriented Lagrangian 3- |
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| ⚫ | In the context of ], a U(N) Chern-Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold N arises as the ] of open strings ending on a ] wrapping M in the ] ] on N. The ] topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern-Simons theory known as ]. | ||
| The information contained in Chern-Simons theory can also be captured through the use of ]. | |||
| ===WZW and matrix models=== | |||
| This theory is ] over boundary manifolds and manifolds with zero ] 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 ] mechanisms. | |||
| Chern-Simons theories are related to many other field theories. For example, if one considers a Chern-Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a 2-dimensional ] known as a G ] on the boundary. In addition the U(N) and SO(N) Chern-Simons theories at large N is well approximated by ]. | |||
| This terms is ] with respect to gauge invariant quantities. | |||
| ==Chern-Simons terms in other theories== | |||
| ⚫ | 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 ] if this term is added to the ]. This term can be induced by integrating over a massive charged ]. It also appears in the ]. | ||
| ⚫ | 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 ] if this term is added to the ]. This term can be induced by integrating over a massive charged ]. It also appears for example in the ]. Ten and eleven dimensional generalizations of Chern-Simons terms appear in the actions of all ten and eleven dimensional ] theories. | ||
| == See also == | == See also == | ||
Revision as of 17:30, 17 August 2006
Chern-Simons theory is a 3-dimensional topological quantum field theory of Schwarz type which computes 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 gauge 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 G 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 action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime.
Relations with other theories
Topological string theories
In the context of string theory, a U(N) Chern-Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold N arises as the string field theory of open strings ending on a D-brane wrapping M in the A-model topological string on N. The B-Model topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern-Simons theory known as holomorphic Chern-Simons theory.
WZW and matrix models
Chern-Simons theories are related to many other field theories. For example, if one considers a Chern-Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a 2-dimensional conformal field theory known as a G Wess-Zumino-Witten model on the boundary. In addition the U(N) and SO(N) Chern-Simons theories at large N is well approximated by matrix models.
Chern-Simons terms in other theories
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 for example in the quantum Hall effect. Ten and eleven dimensional generalizations of Chern-Simons terms appear in the actions of all ten and eleven dimensional supergravity theories.
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.