Editing Chern–Simons theory (section)

===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''.