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Revision as of 20:42, 17 August 2006 editJarahE (talk | contribs)Extended confirmed users523 edits rewording, +quantization← Previous edit Revision as of 22:16, 17 August 2006 edit undoJarahE (talk | contribs)Extended confirmed users523 edits small changes, characterized conformal blocksNext edit →
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In ], Chern-Simons theory is a 3-dimensional ] of ]. It was popularized by ] in 1989, when he demonstrated that it may be used to calculate ] and ] invariants such as the ], as had been conjectured two years earlier by ]. It is named so because its ] is nothing but the integral of the ]. In ], Chern-Simons theory is a 3-dimensional ] of ]. It was popularized by ] in 1989, when he demonstrated that it may be used to calculate ] and ] invariants such as the ], as had been conjectured two years earlier by ]. It is so named because its ] is proportional to the integral of the ].


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


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


===Dynamics=== ===Dynamics===
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where k is the level of the theory. The classical physics of Chern-Simons theory is independent of the choice of level k. 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 the field A. Explicitly the field equation is 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


::<math>0=\frac{\delta S}{\delta A}=\frac{k}{4\pi}\star F</math> ::<math>0=\frac{\delta S}{\delta A}=\frac{k}{4\pi}\star F</math>


where the above star is ]. where <math>\star</math> is ].


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 ] of M to the gauge group G. 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 ] of M to the gauge group G.
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To ] 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 ]. 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 ]s, in fact a state can be defined on any surface. To ] 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 ]. 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 ]s, 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 it can be canonically identified with the space of ]s of the G WZW model at level k. 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. ] 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 ]s of the G WZW model at level k. Conformal blocks are locally ] and antiholomorphic factors whose products sum to the ]s 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 integral ]s of the ] corresponding to g at level k. Characterizations of the conformal blocks at higher ] are not necessary for Witten's solution of Chern-Simons 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 ]s of the ] 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.




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==Chern-Simons terms in other theories== ==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 for example in the ]. Ten and eleven dimensional generalizations of Chern-Simons terms appear in the actions of all ten and eleven dimensional ] 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 action of Maxwell's theory of ]. 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 22:16, 17 August 2006

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 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

S = k 4 π ∫ M T r ( A ∧ d A + 2 3 A ∧ A ∧ A ) {\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

0 = δ S δ A = k 4 π ⋆ F {\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

References

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