Chern–Simons form

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 $\mathbf {A}$ over it, we can define a family of p-forms:^[3]

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{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

\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 F^k. 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.^[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.

References

^ Freed, Daniel (January 15, 2009). "Remarks on Chern–Simons theory" (PDF). Retrieved April 1, 2020.
^ 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.
^ "Chern-Simons form in nLab". ncatlab.org. Retrieved May 1, 2020.
^ Moore, Greg (June 7, 2019). "Introduction To Chern-Simons Theories" (PDF). University of Texas. Retrieved June 7, 2019.
^ 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.

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Chern–Simons form

Definition

Application to physics

See also

References

Further reading