Editing Chern–Simons theory (section)

===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.