Chern–Simons form
Secondary characteristic classes of 3-manifolds
title: "Chern–Simons form" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["homology-theory", "algebraic-topology", "differential-geometry", "string-theory"] description: "Secondary characteristic classes of 3-manifolds" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Chern–Simons_form" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Secondary characteristic classes of 3-manifolds ::
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.
Definition
Given a manifold and a Lie algebra valued 1-form \mathbf{A} over it, we can define a family of p-forms:
In one dimension, the Chern–Simons 1-form is given by :\operatorname{Tr} [ \mathbf{A} ].
In three dimensions, the Chern–Simons 3-form is given by :\operatorname{Tr} \left[ \mathbf{F} \wedge \mathbf{A}-\frac{1}{3} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \right] = \operatorname{Tr} \left[ d\mathbf{A} \wedge \mathbf{A} + \frac{2}{3} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A}\right].
In five dimensions, the Chern–Simons 5-form is given by : \begin{align} & \operatorname{Tr} \left[ \mathbf{F}\wedge\mathbf{F} \wedge \mathbf{A}-\frac{1}{2} \mathbf{F} \wedge\mathbf{A}\wedge\mathbf{A}\wedge\mathbf{A} +\frac{1}{10} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge\mathbf{A} \right] \[6pt] = {} & \operatorname{Tr} \left[ d\mathbf{A}\wedge d\mathbf{A} \wedge \mathbf{A} + \frac{3}{2} d\mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} +\frac{3}{5} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A}\wedge\mathbf{A}\wedge\mathbf{A} \right] \end{align}
where the curvature F is defined as :\mathbf{F} = d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}.
The general Chern–Simons form \omega_{2k-1} is defined in such a way that :d\omega_{2k-1}= \operatorname{Tr}(F^k),
where the wedge product is used to define Fk. The right-hand side of this equation is proportional to the k-th Chern character of the connection \mathbf{A}.
In general, the Chern–Simons p-form is defined for any odd p.
Application to physics
In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.
In the gauge theory, the integral of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.
References
References
- Freed, Daniel. (April 2009). "Remarks on Chern–Simons theory". [[Bulletin of the American Mathematical Society]].
- Chern, S.-S.. (1974). "Characteristic forms and geometric invariants". [[Annals of Mathematics]].
- Chern, Shiing-Shen. (1996). ["A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern"]({{google books). World Scientific.
- "Chern-Simons form in nLab".
- Moore, Greg. (June 7, 2019). "Introduction To Chern-Simons Theories".
- Schwartz, A. S.. (1978). "The partition function of degenerate quadratic functional and Ray-Singer invariants". [[Letters in Mathematical Physics]].
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::