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 [math]\displaystyle{ \mathbf{A} }[/math] over it, we can define a family of p-forms:[3]
In one dimension, the Chern–Simons 1-form is given by
- [math]\displaystyle{ \operatorname{Tr} [ \mathbf{A} ]. }[/math]
In three dimensions, the Chern–Simons 3-form is given by
- [math]\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]. }[/math]
In five dimensions, the Chern–Simons 5-form is given by
- [math]\displaystyle{ \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} }[/math]
where the curvature F is defined as
- [math]\displaystyle{ \mathbf{F} = d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}. }[/math]
The general Chern–Simons form [math]\displaystyle{ \omega_{2k-1} }[/math] is defined in such a way that
- [math]\displaystyle{ d\omega_{2k-1}= \operatorname{Tr}(F^k), }[/math]
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 [math]\displaystyle{ \mathbf{A} }[/math].
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
- ↑ Freed, Daniel (January 15, 2009). "Remarks on Chern–Simons theory". https://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01243-9/S0273-0979-09-01243-9.pdf.
- ↑ Chern, Shiing-Shen; Tian, G.; Li, Peter (1996) (in en). A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern. World Scientific. ISBN 978-981-02-2385-4. https://books.google.com/books?id=uOfSa0sfJr0C&q=Characteristic+Forms+and+Geometric+Invariants&pg=PA363.
- ↑ "Chern-Simons form in nLab". https://ncatlab.org/nlab/show/Chern-Simons+form.
- ↑ Moore, Greg (June 7, 2019). "Introduction To Chern-Simons Theories". http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf.
- ↑ Schwartz, A. S. (1978). "The partition function of degenerate quadratic functional and Ray-Singer invariants". Letters in Mathematical Physics 2 (3): 247–252. doi:10.1007/BF00406412. Bibcode: 1978LMaPh...2..247S.
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.
- 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. https://books.google.com/books?id=FC_DRRUHFXEC&pg=PA321.
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