Chern-Simons functional

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