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]

Definition

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

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

References

  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". ncatlab.org. 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

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