Editing Chern–Simons theory (section)

{{short description|Three-dimensional topological quantum field theory whose action is the Chern–Simons form}}
The '''Chern–Simons theory''' is a 3-dimensional [[topological quantum field theory]] of [[Topological quantum field theory#Schwarz-type TQFTs|Schwarz type]]. It was discovered first by mathematical physicist [[Albert Schwarz]]. It is named after mathematicians [[Shiing-Shen Chern]] and [[James Harris Simons]], who introduced the [[Chern–Simons 3-form]]. In the Chern–Simons theory, the [[action (physics)|action]] is proportional to the integral of the Chern–Simons 3-form.

In [[condensed matter physics|condensed-matter physics]], Chern–Simons theory describes [[Composite fermion|composite fermions]] and the [[topological order]] in [[fractional quantum Hall effect]] states. In mathematics, it has been used to calculate [[knot invariants]] and [[three-manifold]] invariants such as the [[Jones polynomial]].<ref name="wittenjonespolynomial"/>

Particularly, Chern–Simons theory is specified by a choice of simple [[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 (quantum field theory)|partition function]] of the [[quantum field theory|quantum]] theory is [[well-defined]] when the level is an integer and the gauge [[field strength]] vanishes on all [[boundary (topology)|boundaries]] of the 3-dimensional spacetime.

It is also the central mathematical object in theoretical models for [[topological quantum computer]]s (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian [[anyon]]ic model of a TQC, the Yang–Lee–Fibonacci model.<ref name="FK02"/><ref name="WangTQCreview"/>

The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to [[fusion rules]] and [[Virasoro conformal block|conformal blocks]] in [[conformal field theory]], and in particular [[Wess–Zumino–Witten model|WZW theory]].<ref name="wittenjonespolynomial"/><ref name="EMSS89"/>