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In physics, Chern-Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was popularized by Edward Witten in 1989, when he demonstrated that it may be used to calculate knot invariants and three-manifold invariants such as the Jones polynomial, as had been conjectured two years earlier by Albert Schwarz. It is so named because its action is proportional to the integral of 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.

The classical theory

Configurations

Chern-Simons theories can be defined on any topological 3-manifold M, with or without boundary. As these theories are Schwarz-type topological theories, no metric needs to be introduced on M.

Chern-Simons theory is a gauge theory, which means that a classical configuration in the Chern-Simons theory on M with gauge group G is described by a G principle bundle on M. The connection of this bundle is characterized by a connection one-form A which is valued in the Lie algebra g of the Lie group G. In general the connection A is only defined on individual coordinate patches, and the values of A on different patches are related by maps known as gauge transformations. These are characterized by the assertion that the covariant derivative, which is the sum of the exterior derivative operator d and the connection A, transforms in the adjoint representation of the gauge group G. The commutator of the covariant derivative with itself is a g-valued 2-form F called the curvature or field strength and it also transforms in the adjoint representation.

Dynamics

The action S of Chern-Simons theory is proportional to the integral of the Chern-Simons 3-form

${\displaystyle S={\frac {k}{4\pi }}\int _{M}Tr(A\wedge dA+{\frac {2}{3}}A\wedge A\wedge A)}$

where k is the level of the theory. The classical physics of Chern-Simons theory is independent of the choice of level k.

Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the field A. Explicitly the field equation is

${\displaystyle 0={\frac {\delta S}{\delta A}}={\frac {k}{4\pi }}\star F}$

where ${\displaystyle \star }$ is Hodge star.

The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to be flat. Thus the classical solutions to G Chern-Simons theory are the flat connections of G principal bundles on M. Flat connections are determined entirely by holonomies of flat sections around noncontractible cycles on the base M. More precisely, they are in one to one correspondence with maps from the fundamental group of M to the gauge group G.

If M has a boundary N then there is additional data which describes a choice of trivialization of the G principle bundle on N. Such a choice characterizes a map from N to G. The dynamics of this map is described by the Wezz-Zumino-Witten (WZW) model on N at level k.

Quantization

To canonically quantize Chern-Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a Hilbert space. Unlike relativistic quantum field theories, there is no prefered notion of time in a Schwarz-type topological field theory and so one cannot impose that Σ be Cauchy surfaces, in fact a state can be defined on any surface.

Σ is codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. Witten has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k. Conformal blocks are locally holomorphic and antiholomorphic factors whose products sum to the correlation functions of a 2-dimensional conformal field theory.

For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern-Simons theory.

Relationships 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 action of Maxwell's theory of electrodynamics. 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

• Chern-Simons form
• topological quantum field theory

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.

• Quantum field theory