Chern–Simons form

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In mathematics, the Chern–Simons forms are certain secondary characteristic classes.[1] 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.[2]


Given a manifold and a Lie algebra valued 1-form over it, we can define a family of p-forms:[3]

In one dimension, the Chern–Simons 1-form is given by

In three dimensions, the Chern–Simons 3-form is given by

In five dimensions, the Chern–Simons 5-form is given by

where the curvature F is defined as

The general Chern–Simons form is defined in such a way that

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 .

In general, the Chern–Simons p-form is defined for any odd p.[4]

Application to physics[edit]

In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.[5]

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.

See also[edit]


  1. ^ Freed, Daniel (January 15, 2009). "Remarks on Chern–Simons theory" (PDF). Retrieved April 1, 2020.
  2. ^ Chern, Shiing-Shen; Tian, G.; Li, Peter (1996). A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern. World Scientific. ISBN 978-981-02-2385-4.
  3. ^ "Chern-Simons form in nLab". Retrieved May 1, 2020.
  4. ^ Moore, Greg (June 7, 2019). "Introduction To Chern-Simons Theories" (PDF). University of Texas. Retrieved June 7, 2019.
  5. ^ Schwartz, A. S. (1978). "The partition function of degenerate quadratic functional and Ray-Singer invariants". Letters in Mathematical Physics. 2 (3): 247–252. Bibcode:1978LMaPh...2..247S. doi:10.1007/BF00406412. S2CID 123231019.

Further reading[edit]