The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. 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 is proportional to the integral of the Chern–Simons 3-form.
In condensed-matter physics, Chern–Simons theory describes 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.
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 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.
It is also the central mathematical object in theoretical models for topological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model.
The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory, and in particular WZW theory.
The classical theory
Mathematical origin
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. Given a flat G-principal bundle P on M there exists a unique homomorphism, called the Chern–Weil homomorphism, from the algebra of G-adjoint invariant polynomials on g (Lie algebra of G) to the cohomology . If the invariant polynomial is homogeneous one can write down concretely any k-form of the closed connection ω as some 2k-form of the associated curvature form Ω of ω.
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. This form is called Chern–Simons form. If df(ω) is closed one can integrate the above formula
where C is a (2k − 1)-dimensional cycle on M. This invariant is called Chern–Simons invariant. 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. It also can be defined as
where 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.
The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The action integral (path integral) of the field theory in physics is viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on M. These explain why the Chern–Simons theory is closely related to topological field 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 characterized 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 values 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 connection A, transforms in the 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.
Dynamics
The action S of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form
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.