Chern–Simons theory
The Chern-Simons theory is a 3-dimension-
al topological quantum field theory of Sch-
warz type, developed by Shiing-Shen Chern
and James Harris Simons. In condensed mat-
ter physics, Chern-Simons theory describes
the topological order in fractional quantum
Hall effect states. It was popularized by Ed-
ward Witten in 1989, when he demonstrated
that it may be used to calculate knot invari-
ants and three-manifold invariants such as
the Jones polynomial, as had been conjec-
tured 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 spe-
cified 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 principal G-bundle
on M. The connection of this bundle is char-
acterized 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 val-
ues 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 con-
nection A, transforms in the adjoint repres-
entation of the gauge group G. The square of
the covariant derivative with itself