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A [[Lagrangian|Lagrangian]] in the theory of gauge fields on an oriented [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201401.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201402.png" />. More precisely, it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201403.png" />-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201404.png" /> on the space of connections ( "gauge fields" ) on a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201406.png" />-bundle (cf. also [[Principal G-object|Principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201407.png" />-object]]) with base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201408.png" /> for a compact connected [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c1201409.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014010.png" /> simply connected, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014011.png" />, the bundle can be taken to be the product bundle and the Chern–Simons functional is given by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014012.png" /></td> </tr></table>
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where the connection is given by the matrix-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014013.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014015.png" /> is the usual trace of matrices (cf. also [[Trace of a square matrix|Trace of a square matrix]]).
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A [[Lagrangian|Lagrangian]] in the theory of gauge fields on an oriented [[Manifold|manifold]] $M$ of dimension $3$. More precisely, it is a ${\bf R} / 2 \pi \bf Z$-valued function $\operatorname{CS}$ on the space of connections ( "gauge fields" ) on a principal $G$-bundle (cf. also [[Principal G-object|Principal $G$-object]]) with base space $M$ for a compact connected [[Lie group|Lie group]] $G$. For $G$ simply connected, e.g. $G = \operatorname {SU} ( N )$, the bundle can be taken to be the product bundle and the Chern–Simons functional is given by the formula
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014016.png" /> is invariant under gauge transformations, i.e. automorphisms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014017.png" />-bundle, and hence it defines a Lagrangian on the space of orbits for the action of the group of these. The critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014018.png" /> are given by the flat connections, i.e. those for which the [[Curvature|curvature]]
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\begin{equation*} \operatorname {CS} ( A ) = \frac { 1 } { 4 \pi } \int _ { M } \operatorname { Tr } ( A \bigwedge d A + \frac { 2 } { 3 } A \bigwedge A \bigwedge A ) \operatorname { mod } 2 \pi , \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014019.png" /></td> </tr></table>
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where the connection is given by the matrix-valued $1$-form $A$ and $\operatorname { Tr}$ is the usual trace of matrices (cf. also [[Trace of a square matrix|Trace of a square matrix]]).
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$\operatorname{CS}$ is invariant under gauge transformations, i.e. automorphisms of the $G$-bundle, and hence it defines a Lagrangian on the space of orbits for the action of the group of these. The critical points of $\operatorname{CS}$ are given by the flat connections, i.e. those for which the [[Curvature|curvature]]
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\begin{equation*} F _ { A } = d A + A \bigwedge A \end{equation*}
  
 
vanishes (cf. also [[Connection|Connection]]).
 
vanishes (cf. also [[Connection|Connection]]).
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1) Using the Chern–Simons functional as a [[Morse function|Morse function]], A. Flöer [[#References|[a6]]] defined invariants for homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014020.png" />-spheres related to the Casson invariant (see [[#References|[a7]]]).
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1) Using the Chern–Simons functional as a [[Morse function|Morse function]], A. Flöer [[#References|[a6]]] defined invariants for homology $3$-spheres related to the Casson invariant (see [[#References|[a7]]]).
  
2) E. Witten [[#References|[a8]]] used the Chern–Simons functional to set up a topological quantum field theory (cf. also [[Quantum field theory|Quantum field theory]]), which gives rise to invariants for knots and links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014022.png" />-manifolds including the Jones polynomial for knots in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120140/c12014023.png" />-sphere. See also [[#References|[a1]]] and [[#References|[a2]]].
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2) E. Witten [[#References|[a8]]] used the Chern–Simons functional to set up a topological quantum field theory (cf. also [[Quantum field theory|Quantum field theory]]), which gives rise to invariants for knots and links in $3$-manifolds including the Jones polynomial for knots in the $3$-sphere. See also [[#References|[a1]]] and [[#References|[a2]]].
  
 
The Chern–Simons functional is a special case of the Chern–Simons invariant and characteristic classes. General references are [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]].
 
The Chern–Simons functional is a special case of the Chern–Simons invariant and characteristic classes. General references are [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Axelrod,  I.M. Singer,  "Chern–Simons perturbation theory" , ''Proc. XXth Internat. Conf. on Differential Geometric Methods in Theoretical Physics (New York, 1991)'' , World Sci.  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Axelrod,  I.M. Singer,  "Chern–Simons pertubation theory II"  ''J. Diff. Geom.'' , '''39'''  (1994)  pp. 173–213</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Cheeger,  J. Simons,  "Differential characters and geometric invariants" , ''Geometry and Topology (Maryland, 1983/4'' , ''Lecture Notes Math.'' , '''1167''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.-S. Chern,  J. Simons,  "Characteristic forms and geometric invariants"  ''Ann. of Math.'' , '''99'''  (1974)  pp. 48–69</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.L. Dupont,  F.W. Kamber,  "On a generalization of Cheeger–Chern–Simons classes"  ''Illinois J. Math.'' , '''33'''  (1990)  pp. 221–255</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Floer,  "An instanton-invariant for 3-manifolds"  ''Comm. Math. Phys.'' , '''118'''  (1988)  pp. 215–240</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C.H. Taubes,  "Casson's invariant and gauge theory"  ''J. Diff. Geom.'' , '''31'''  (1990)  pp. 547–599</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Witten,  "Quantum field theory and the Jones polynomial"  ''Comm. Math. Phys.'' , '''121'''  (1989)  pp. 351–399</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  S. Axelrod,  I.M. Singer,  "Chern–Simons perturbation theory" , ''Proc. XXth Internat. Conf. on Differential Geometric Methods in Theoretical Physics (New York, 1991)'' , World Sci.  (1992)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Axelrod,  I.M. Singer,  "Chern–Simons pertubation theory II"  ''J. Diff. Geom.'' , '''39'''  (1994)  pp. 173–213</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Cheeger,  J. Simons,  "Differential characters and geometric invariants" , ''Geometry and Topology (Maryland, 1983/4'' , ''Lecture Notes Math.'' , '''1167''' , Springer  (1985)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  S.-S. Chern,  J. Simons,  "Characteristic forms and geometric invariants"  ''Ann. of Math.'' , '''99'''  (1974)  pp. 48–69</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J.L. Dupont,  F.W. Kamber,  "On a generalization of Cheeger–Chern–Simons classes"  ''Illinois J. Math.'' , '''33'''  (1990)  pp. 221–255</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Floer,  "An instanton-invariant for 3-manifolds"  ''Comm. Math. Phys.'' , '''118'''  (1988)  pp. 215–240</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  C.H. Taubes,  "Casson's invariant and gauge theory"  ''J. Diff. Geom.'' , '''31'''  (1990)  pp. 547–599</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E. Witten,  "Quantum field theory and the Jones polynomial"  ''Comm. Math. Phys.'' , '''121'''  (1989)  pp. 351–399</td></tr></table>

Latest revision as of 16:58, 1 July 2020

A Lagrangian in the theory of gauge fields on an oriented manifold $M$ of dimension $3$. More precisely, it is a ${\bf R} / 2 \pi \bf Z$-valued function $\operatorname{CS}$ on the space of connections ( "gauge fields" ) on a principal $G$-bundle (cf. also Principal $G$-object) with base space $M$ for a compact connected Lie group $G$. For $G$ simply connected, e.g. $G = \operatorname {SU} ( N )$, the bundle can be taken to be the product bundle and the Chern–Simons functional is given by the formula

\begin{equation*} \operatorname {CS} ( A ) = \frac { 1 } { 4 \pi } \int _ { M } \operatorname { Tr } ( A \bigwedge d A + \frac { 2 } { 3 } A \bigwedge A \bigwedge A ) \operatorname { mod } 2 \pi , \end{equation*}

where the connection is given by the matrix-valued $1$-form $A$ and $\operatorname { Tr}$ is the usual trace of matrices (cf. also Trace of a square matrix).

$\operatorname{CS}$ is invariant under gauge transformations, i.e. automorphisms of the $G$-bundle, and hence it defines a Lagrangian on the space of orbits for the action of the group of these. The critical points of $\operatorname{CS}$ are given by the flat connections, i.e. those for which the curvature

\begin{equation*} F _ { A } = d A + A \bigwedge A \end{equation*}

vanishes (cf. also Connection).

Applications of the Chern–Simons functional.

1) Using the Chern–Simons functional as a Morse function, A. Flöer [a6] defined invariants for homology $3$-spheres related to the Casson invariant (see [a7]).

2) E. Witten [a8] used the Chern–Simons functional to set up a topological quantum field theory (cf. also Quantum field theory), which gives rise to invariants for knots and links in $3$-manifolds including the Jones polynomial for knots in the $3$-sphere. See also [a1] and [a2].

The Chern–Simons functional is a special case of the Chern–Simons invariant and characteristic classes. General references are [a3], [a4], [a5].

References

[a1] S. Axelrod, I.M. Singer, "Chern–Simons perturbation theory" , Proc. XXth Internat. Conf. on Differential Geometric Methods in Theoretical Physics (New York, 1991) , World Sci. (1992)
[a2] S. Axelrod, I.M. Singer, "Chern–Simons pertubation theory II" J. Diff. Geom. , 39 (1994) pp. 173–213
[a3] J. Cheeger, J. Simons, "Differential characters and geometric invariants" , Geometry and Topology (Maryland, 1983/4 , Lecture Notes Math. , 1167 , Springer (1985)
[a4] S.-S. Chern, J. Simons, "Characteristic forms and geometric invariants" Ann. of Math. , 99 (1974) pp. 48–69
[a5] J.L. Dupont, F.W. Kamber, "On a generalization of Cheeger–Chern–Simons classes" Illinois J. Math. , 33 (1990) pp. 221–255
[a6] A. Floer, "An instanton-invariant for 3-manifolds" Comm. Math. Phys. , 118 (1988) pp. 215–240
[a7] C.H. Taubes, "Casson's invariant and gauge theory" J. Diff. Geom. , 31 (1990) pp. 547–599
[a8] E. Witten, "Quantum field theory and the Jones polynomial" Comm. Math. Phys. , 121 (1989) pp. 351–399
How to Cite This Entry:
Chern-Simons functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern-Simons_functional&oldid=22287
This article was adapted from an original article by J.L. Dupont (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article