Difference between revisions of "Chern-Simons functional"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (moved Chern–Simons functional to Chern-Simons functional: ascii title) |
(No difference)
|
Revision as of 18:51, 24 March 2012
A Lagrangian in the theory of gauge fields on an oriented manifold of dimension
. More precisely, it is a
-valued function
on the space of connections ( "gauge fields" ) on a principal
-bundle (cf. also Principal
-object) with base space
for a compact connected Lie group
. For
simply connected, e.g.
, the bundle can be taken to be the product bundle and the Chern–Simons functional is given by the formula
![]() |
where the connection is given by the matrix-valued -form
and
is the usual trace of matrices (cf. also Trace of a square matrix).
is invariant under gauge transformations, i.e. automorphisms of the
-bundle, and hence it defines a Lagrangian on the space of orbits for the action of the group of these. The critical points of
are given by the flat connections, i.e. those for which the curvature
![]() |
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 -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 -manifolds including the Jones polynomial for knots in the
-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 |
Chern-Simons functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern-Simons_functional&oldid=22287