It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form.
In condensed-matter physics, Chern–Simons theory describes the topological order in fractional quantum Hall effect states.
It is also the central mathematical object in theoretical models for topological quantum computers (TQC).
[1][4] In the 1940s S. S. Chern and 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.
In 1974 S. S. Chern and J. H. Simons had concretely constructed a (2k − 1)-form df(ω) such that where T is the Chern–Weil homomorphism.
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.
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.
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 (WZW) model on N at level k. To canonically quantize Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a Hilbert space.
After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model.
When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.
In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known knot polynomials.
For example, in G = U(N) Chern–Simons theory at level k the normalized correlation function is, up to a phase, equal to times the HOMFLY polynomial.
The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data.
This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzero normal vector at each point along which one deforms the loop to calculate its self-linking number.
Sir Michael Atiyah has shown that there exists a canonical choice of 2-framing,[5] which is generally used in the literature today and leads to a well-defined linking number.
Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R3).
For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a two-dimensional conformal field theory known as a G Wess–Zumino–Witten model on the boundary.
In three dimensions, the gravitational Chern–Simons term is This variation gives the Cotton tensor Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action.
In 2013 Kenneth A. Intriligator and Nathan Seiberg solved these 3d Chern–Simons gauge theories and their phases using monopoles carrying extra degrees of freedom.
In 2013 Kevin Costello defined a closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve.
The Chern–Simons term can also be added to models which aren't topological quantum field theories.
In 3D, this gives rise to a massive photon if this term is added to the action of Maxwell's theory of electrodynamics.
This term can be induced by integrating over a massive charged Dirac field.
The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions[12][13] Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional supergravity theories.