Chern–Simons form

Given a manifold and a Lie algebra valued 1-form $\backslash mathbf\{A\}$ over it, we can define a family of p-forms:{{Cite web|title=Chern-Simons form in nLab|url=https://ncatlab.org/nlab/show/Chern-Simons+form|website=ncatlab.org|access-date=May 1, 2020}}

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

:$\backslash operatorname\{Tr\}\; [\; \backslash mathbf\{A\}\; ].$

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

:$\backslash operatorname\{Tr\}\; \backslash left[\; \backslash mathbf\{F\}\; \backslash wedge\; \backslash mathbf\{A\}-\backslash frac\{1\}\{3\}\; \backslash mathbf\{A\}\; \backslash wedge\; \backslash mathbf\{A\}\; \backslash wedge\; \backslash mathbf\{A\}\; \backslash right]\; =\; \backslash operatorname\{Tr\}\; \backslash left[\; d\backslash mathbf\{A\}\; \backslash wedge\; \backslash mathbf\{A\}\; +\; \backslash frac\{2\}\{3\}\; \backslash mathbf\{A\}\; \backslash wedge\; \backslash mathbf\{A\}\; \backslash wedge\; \backslash mathbf\{A\}\backslash right].$

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

:$\backslash mathbf\{F\}\; =\; d\backslash mathbf\{A\}+\backslash mathbf\{A\}\backslash wedge\backslash mathbf\{A\}.$

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

:$d\backslash omega\_\{2k-1\}=\; \backslash operatorname\{Tr\}(F^k),$

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 $\backslash mathbf\{A\}$.

In general, the Chern–Simons p-form is defined for any odd p.{{Cite web|title=Introduction To Chern-Simons Theories|url=http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf|last=Moore|first=Greg|date=June 7, 2019|website=University of Texas|access-date=June 7, 2019}}

In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.{{cite journal |last=Schwartz |first=A. S. |year=1978 |title=The partition function of degenerate quadratic functional and Ray-Singer invariants |journal=Letters in Mathematical Physics |volume=2 |issue=3 |pages=247–252 |doi=10.1007/BF00406412 |bibcode=1978LMaPh...2..247S |s2cid=123231019 }}

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.

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

{{Reflist}}

• {{cite journal |last1=Chern |first1=S.-S. |author-link1=Shiing-Shen Chern |last2=Simons |first2=J. |author-link2=James Harris Simons |title=Characteristic forms and geometric invariants |journal=Annals of Mathematics |series=Second Series |volume=99 |number=1 |year=1974 |pages=48–69 |jstor=1971013 |doi=10.2307/1971013 }}
• {{cite book |first=Reinhold A. |last=Bertlmann |chapter=Chern–Simons form, homotopy operator and anomaly |title=Anomalies in Quantum Field Theory |publisher=Clarendon Press |year=2001 |edition=Revised |pages=321–341 |isbn=0-19-850762-3 |chapter-url=https://books.google.com/books?id=FC_DRRUHFXEC&pg=PA321 }}

{{String theory topics |state=collapsed}}

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