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 $𝐀$ over it, we can define a family of p-forms:^[4]

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

Tr [𝐀] .

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

Tr [𝐅 \land 𝐀 - \frac{1}{3} 𝐀 \land 𝐀 \land 𝐀] = Tr [d 𝐀 \land 𝐀 + \frac{2}{3} 𝐀 \land 𝐀 \land 𝐀] .

In five dimensions, the Chern–Simons 5-form is given by

\begin{matrix} Tr [𝐅 \land 𝐅 \land 𝐀 - \frac{1}{2} 𝐅 \land 𝐀 \land 𝐀 \land 𝐀 + \frac{1}{10} 𝐀 \land 𝐀 \land 𝐀 \land 𝐀 \land 𝐀] \\ = & Tr [d 𝐀 \land d 𝐀 \land 𝐀 + \frac{3}{2} d 𝐀 \land 𝐀 \land 𝐀 \land 𝐀 + \frac{3}{5} 𝐀 \land 𝐀 \land 𝐀 \land 𝐀 \land 𝐀] \end{matrix}

where the curvature F is defined as

𝐅 = d 𝐀 + 𝐀 \land 𝐀 .

The general Chern–Simons form $ω_{2 k - 1}$ is defined in such a way that

d ω_{2 k - 1} = 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 $𝐀$ .

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.

References

^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

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Chern–Simons form

Contents

Definition

Application to physics

See also

References

Further reading

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Chern–Simons form

Definition

Application to physics

See also

References

Further reading

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