Chern-Simons form

A Chern-Simons form is an odd-degree transgression form whose exterior derivative is an invariant polynomial in the curvature form. For the trace polynomial of degree $k$, $d{\omega }_{2k-1}=\mathrm{Tr}(F^k)$; the Chern-Simons 3-form is the degree-three case.

The Chern-Simons 3-formsatisfies $dY=C_2$.

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.

For an anti-Hermitian $SU(N)$ connection one-form, normalize the Chern-Simons three-form by $1/(8{\pi }^2)$ and use $A^g=g^{-1}Ag+g^{-1}dg$. Its change is $-d\mathrm{Tr}(dg{g}^{-1}\wedge A)/(8{\pi }^2)-\mathrm{Tr}(g^{-1}dg{)}^3/(24{\pi }^2)$. On a closed three-manifold the first term integrates to zero, while the latter is an integer with the fundamental matrix trace. Hence the integral is defined modulo integers; integer-level exponentials are invariant under large Yang-Mills gauge transformations.