= A-hat form
{title2=$\widehat A(R)=\det{}^{1/2}\!\left(\frac{R/(4\pi i)}{\sinh(R/(4\pi i))}\right)$}
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 $p_1=-\operatorname{tr}(R\wedge R)/(8\pi^2)$ is the first <Pontryagin class> form.
Back to article page
