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Chern–Simons theory
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Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of 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 is proportional to the integral of the Chern–Simons 3-form.
In condensed-matter physics, Chern–Simons theory describes 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.
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.
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
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Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of 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 is proportional to the integral of the Chern–Simons 3-form.
In condensed-matter physics, Chern–Simons theory describes 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.
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.
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