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Definition

Given a manifold and a Lie algebra valued 1-form A {\displaystyle \mathbf {A} } over it, we can define a family of p-forms:³

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

Tr ⁡ [ A ] . {\displaystyle \operatorname {Tr} [\mathbf {A} ].}

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

Tr ⁡ [ F ∧ A − 1 3 A ∧ A ∧ A ] = Tr ⁡ [ d A ∧ A + 2 3 A ∧ A ∧ A ] . {\displaystyle \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

Tr ⁡ [ F ∧ F ∧ A − 1 2 F ∧ A ∧ A ∧ A + 1 10 A ∧ A ∧ A ∧ A ∧ A ] = Tr ⁡ [ d A ∧ d A ∧ A + 3 2 d A ∧ A ∧ A ∧ A + 3 5 A ∧ A ∧ A ∧ A ∧ A ] {\displaystyle {\begin{aligned}&\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{aligned}}}

where the curvature F is defined as

F = d A + A ∧ A . {\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}

The general Chern–Simons form ω 2 k − 1 {\displaystyle \omega _{2k-1}} is defined in such a way that

d ω 2 k − 1 = Tr ⁡ ( F k ) , {\displaystyle 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 A {\displaystyle \mathbf {A} } .

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

• Chern–Weil homomorphism
• Chiral anomaly
• Topological quantum field theory
• Jones polynomial

Further reading

• Chern, S.-S.; Simons, J. (1974). "Characteristic forms and geometric invariants". Annals of Mathematics. Second Series. 99 (1): 48–69. doi:10.2307/1971013. JSTOR 1971013.
• Bertlmann, Reinhold A. (2001). "Chern–Simons form, homotopy operator and anomaly". Anomalies in Quantum Field Theory (Revised ed.). Clarendon Press. pp. 321–341. ISBN 0-19-850762-3.

References

1. Freed, Daniel (January 15, 2009). "Remarks on Chern–Simons theory" (PDF). Retrieved April 1, 2020. https://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01243-9/S0273-0979-09-01243-9.pdf ↩

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. 978-981-02-2385-4 ↩

3. "Chern-Simons form in nLab". ncatlab.org. Retrieved May 1, 2020. https://ncatlab.org/nlab/show/Chern-Simons+form ↩

4. Moore, Greg (June 7, 2019). "Introduction To Chern-Simons Theories" (PDF). University of Texas. Retrieved June 7, 2019. http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf ↩

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. /wiki/Bibcode_(identifier) ↩