Editing Chern–Simons theory (section)

===Mathematical origin===

In the 1940s [[Shiing-Shen Chern|S. S. Chern]] and [[André Weil|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 <math>H^*(M,\mathbb{R})</math>. If the invariant polynomial is homogeneous one can write down concretely any ''k''-form of the closed connection ''ω'' as some 2''k''-form of the associated curvature form Ω of ''ω''.

In 1974 S. S. Chern and [[James Harris Simons|J. H. Simons]] had concretely constructed a (2''k''&nbsp;&minus;&nbsp;1)-form ''df''(''ω'') such that
:<math>dTf(\omega)=f(\Omega^k),</math>
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
:<math>Tf(\omega)=\int_C f(\Omega^k),</math>
where ''C'' is a (2''k''&nbsp;&minus;&nbsp;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
:<math>\operatorname{CS}(M)=\int_{s(M)}\tfrac{1}{2}Tp_1\in\mathbb{R}/\mathbb{Z},</math>
where <math>p_1</math> 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 formulation|path integral]]) of the [[quantum field theory|field theory]] in physics is viewed as the [[Lagrangian (field theory)|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]].