Editing Chern–Simons theory (section)

===Chern–Simons gravity theory===
{{See also|(2+1)-dimensional topological gravity}}
In 1982, [[Stanley Deser|S. Deser]], [[Roman Jackiw|R. Jackiw]] and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the [[Einstein–Hilbert action]] in gravity theory is modified by adding the Chern–Simons term. ({{harvtxt|Deser|Jackiw|Templeton|1982}})

In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions ({{harvtxt|Jackiw|Pi|2003}}) and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.

The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is
:<math>\operatorname{CS}(\Gamma)=\frac{1}{2\pi^2}\int d^3x\varepsilon^{ijk}\biggl(\Gamma^p_{iq}\partial_j\Gamma^q_{kp}+\frac{2}{3}\Gamma^p_{iq}\Gamma^q_{jr}\Gamma^r_{kp}\biggr).</math>
This variation gives the [[Cotton tensor]]
:<math>=-\frac{1}{2\sqrt{g}}\bigl(\varepsilon^{mij}D_i R^n_j+\varepsilon^{nij}D_i R^m_j).</math>

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.