Chern-Simons number of a gauge field ($N_{\mathrm{CS}}$)

For an oriented three-dimensional slice $\Sigma$, use the Skew-Hermitian trace conventionSince the differential of the integrand is $\mathrm{tr}(F\wedge F)$, a space-time slab oriented by $dt\wedge {\mathrm{vol}}_{\Sigma }$ has instanton charge $k=N_{\mathrm{CS}}(t_{+})-N_{\mathrm{CS}}(t_{-})$ when the side boundary contributes no flux. For a pure gauge $A=h^{-1}dh$, this reduces to the winding integral $(24{\pi }^2{)}^{-1}{\int }_{\Sigma }\mathrm{tr}(h^{-1}dh{)}^3$. Gauge transformations of nonzero winding can shift it by an integer.