= Chern-Simons number of a gauge field
{c}
{title2=$N_{\rm CS}$}
= Chern-Simons number
{c}
{synonym}
For an oriented three-dimensional slice $\Sigma$, use the <Skew-Hermitian> trace convention
$$
N_{\rm CS}[A]= -\frac1{8\pi^2}\int_\Sigma\operatorname{tr}\left(A\wedge dA+\frac23A\wedge A\wedge A\right).
$$
Since the differential of the integrand is $\operatorname{tr}(F\wedge F)$, a space-time slab oriented by $dt\wedge\operatorname{vol}_\Sigma$ has <instanton> charge $k=N_{\rm CS}(t_+)-N_{\rm CS}(t_-)$ when the side boundary contributes no flux. For a pure gauge $A=h^{-1}dh$, this reduces to the winding integral $(24\pi^2)^{-1}\int_\Sigma\operatorname{tr}(h^{-1}dh)^3$. <Gauge transformations> of nonzero winding can shift it by an <integer>.
Back to article page
