Editing Chern–Simons theory (section)

==The classical theory==

===Mathematical origin===

In the 1940s [[Shiing-Shen Chern|S. S. Chern]] and [[André Weil|A. Weil]] studied the global curvature properties of smooth manifolds ''M'' as [[de Rham cohomology]] ([[Chern–Weil theory]]), which is an important step in the theory of [[characteristic classes]] in [[differential geometry]]. Given a flat ''G''-[[principal bundle]] ''P'' on ''M'' there exists a unique homomorphism, called the [[Chern–Weil homomorphism]], from the algebra of ''G''-adjoint invariant polynomials on ''g'' (Lie algebra of ''G'') to the cohomology <math>H^*(M,\mathbb{R})</math>. If the invariant polynomial is homogeneous one can write down concretely any ''k''-form of the closed connection ''ω'' as some 2''k''-form of the associated curvature form Ω of ''ω''.

In 1974 S. S. Chern and [[James Harris Simons|J. H. Simons]] had concretely constructed a (2''k''&nbsp;&minus;&nbsp;1)-form ''df''(''ω'') such that
:<math>dTf(\omega)=f(\Omega^k),</math>
where ''T'' is the Chern–Weil homomorphism. This form is called [[Chern–Simons form]]. If ''df''(''ω'') is closed one can integrate the above formula
:<math>Tf(\omega)=\int_C f(\Omega^k),</math>
where ''C'' is a (2''k''&nbsp;&minus;&nbsp;1)-dimensional cycle on ''M''. This invariant is called '''Chern–Simons invariant'''. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(''M'') is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as
:<math>\operatorname{CS}(M)=\int_{s(M)}\tfrac{1}{2}Tp_1\in\mathbb{R}/\mathbb{Z},</math>
where <math>p_1</math> is the first Pontryagin number and ''s''(''M'') is the section of the normal orthogonal bundle ''P''. Moreover, the Chern–Simons term is described as the [[eta invariant]] defined by Atiyah, Patodi and Singer.

The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The [[action integral]] ([[Path integral formulation|path integral]]) of the [[quantum field theory|field theory]] in physics is viewed as the [[Lagrangian (field theory)|Lagrangian]] integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on ''M''. These explain why the Chern–Simons theory is closely related to [[topological field theory]].

===Configurations===

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

Chern–Simons theory is a [[gauge theory]], which means that a [[classical physics|classical]] configuration in the Chern–Simons theory on ''M'' with [[gauge group]] ''G'' is described by a [[principal bundle|principal ''G''-bundle]] on ''M''. The [[connection (principal bundle)|connection]] of this bundle is characterized by a [[connection one-form]] ''A'' which is [[vector-valued differential form#Lie algebra-valued forms|valued]] in the [[Lie algebra]] '''g''' of the [[Lie group]] ''G''. In general the connection ''A'' is only defined on individual [[coordinate patch]]es, and the values of ''A'' on different patches are related by maps known as [[gauge symmetry|gauge transformations]]. These are characterized by the assertion that the [[gauge covariant derivative|covariant derivative]], which is the sum of the [[exterior derivative]] operator ''d'' and the connection ''A'', transforms in the [[Adjoint representation of a Lie group|adjoint representation]] of the gauge group ''G''. The square of the covariant derivative with itself can be interpreted as a '''g'''-valued 2-form ''F'' called the [[curvature form]] or [[field strength]]. It also transforms in the adjoint representation.

===Dynamics===
The [[action (physics)|action]] ''S'' of Chern–Simons theory is proportional to the integral of the [[Chern–Simons 3-form]]

:<math>S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A).</math>

The constant ''k'' is called 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''. In terms of the field curvature

:<math>F = dA + A \wedge A \, </math>

the [[field equation]] is explicitly

:<math>0=\frac{\delta S}{\delta A}=\frac{k}{2\pi} F.</math>

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 principal ''G''-bundles on ''M''. Flat connections are determined entirely by holonomies around noncontractible cycles on the base ''M''. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the [[fundamental group]] of ''M'' to the gauge group ''G'' up to conjugation.

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