Instanton number ($k$)

For an $SU(2)$ connection under a standard trace convention,Its sign depends on orientation and on whether the instanton is self-dual or anti-self-dual.

For a finite-action $SU(2)$ connection on ${\mathbb{R}}^4$ that approaches $A=-dg{g}^{-1}$ at infinity,under the convention $k=(8{\pi }^2{)}^{-1}\int \mathrm{Tr}(F\wedge F)$. This integer is the degree of a map between oriented manifolds $g:S_{\infty }^3\to SU(2)\cong S^3$; orientation and trace conventions may reverse its sign.