View source for Chern-Simons functional

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

<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>

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]]).

<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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vanishes (cf. also [[Connection|Connection]]).

==Applications of the Chern–Simons functional.==


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]]]).

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]]].

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====
<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>