Chern-Weil homomorphism

Chern-Weil theory constructs closed characteristic forms by applying invariant polynomials to the curvature of a connection.

For a real vector bundle $E$, its Pontryagin classes are the even-index Chern classes of its complexification with the displayed alternating signs. They lie in integral cohomology of degrees $4k$ and obey a total-class product formula under direct sums. If a metric connection on a vector bundle has curvature matrix $R$, its first real representative is $p_1(E)=-\mathrm{tr}(R\wedge R)/(8{\pi }^2)$. The Bianchi identity makes this representative closed, and Chern-Weil connection transgression makes its de Rham cohomology class independent of the connection. This normalization gives degree-four A-hat form $-p_1/24$.

For the vector-bundle curvature matrix $R$ of a metric connection on a vector bundle on a real oriented vector bundle, the A-hat form is the even differential form defined by the displayed finite-degree power series, with square root having constant term one. The expression is independent of the choice of orthonormal frame. Equivalently, for formal paired curvature Chern roots $\pm x_j$, it is ${\prod }_j(x_j/2)/\sinh (x_j/2)$. The Bianchi identity makes it closed, and changing the connection changes it by an exact differential form via Chern-Weil connection transgression. Its degree-four part is $-p_1/24$, where $p_1=-\mathrm{tr}(R\wedge R)/(8{\pi }^2)$ is the first Pontryagin class form.

A variation of the connection one-form has $\dot{F}=D_A\dot{A}$. The Bianchi identity and graded cyclic matrix trace imply ${\partial }_t\mathrm{Tr}(F\wedge F)=2d\mathrm{Tr}(\dot{A}\wedge F)$. Thus the Chern-Weil theory four-form changes by an exact differential form, and its integral on a closed four-cycle is connection-independent. For a general unitary bundle, include the corresponding trace-product term in the Second Chern form.

For curvature $F=dA+A\wedge A$, a standard normalization of the second Chern form is

The second Chern number is the integral of the second Chern class over an oriented four-cycle. For a suitably normalized $SU(n)$ curvature on a compactified four-manifold, it is an integer.

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.