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][3]

Definition

Given a manifold and a Lie algebra valued 1-form \mathbf{A} over it, we can define a family of p-forms:^[4]

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{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}

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.^[5]

Application to physics

In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.^[6]

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

References

1. ^ Freed, Daniel (April 2009 [January 15, 2009]). "Remarks on Chern–Simons theory". Bulletin of the American Mathematical Society. 46 (2): 221–254. Retrieved April 1, 2020.
2. ^ Chern, S.-S. & Simons, J. (1974). "Characteristic forms and geometric invariants". Annals of Mathematics. 99 (1): 48–69. Second Series. doi:10.2307/1971013. JSTOR 1971013
3. ^ 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.
4. ^ "Chern-Simons form in nLab". ncatlab.org. Retrieved May 1, 2020.
5. ^ Moore, Greg (June 7, 2019). "Introduction To Chern-Simons Theories". University of Texas. Retrieved June 7, 2019.
6. ^ 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

• 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.