BPST instanton

Remarks

Check that it is an instanton: expanding in coordinates, the $i$-component of $d\bar x \wedge dx$ is $2(dx_0 \wedge dx_1 - dx_2 \wedge dx_3)$, and the $j$- and $k$-components are $2(dx_0 \wedge dx_2 - dx_3 \wedge dx_1)$ and $2(dx_0 \wedge dx_3 - dx_1 \wedge dx_2)$. Each of these $2$-forms is anti-self-dual for the Hodge star of the orientation $dx_0 \wedge dx_1 \wedge dx_2 \wedge dx_3$, since $\star(dx_0 \wedge dx_1) = dx_2 \wedge dx_3$ and so on. So $\star F_A = -F_A$.

The curvature decays like $\abs{x}^{-4}$, so the Yang–Mills action $\int_{\R^4} \abs{F_A}^2$ is finite. By conformal invariance the connection extends to a bundle over the sphere $S^4$, and the instanton number of that bundle has absolute value $1$. Up to gauge transformations, the parameters $(a, \lambda) \in \R^4 \times (0, \infty)$ describe all anti-self-dual connections of instanton number $1$ for this orientation. The moduli space of these connections over $S^4$ is an open $5$-dimensional ball. The scale $\lambda$ measures the size of the region where the curvature is concentrated; as $\lambda \to 0$ the curvature concentrates at the point $a$.

Belavin, Polyakov, Schwartz and Tyupkin found this solution in 1975, and the name is formed from their initials. In quantum field theory it describes tunnelling between vacua of $SU(2)$ Yang–Mills theory in Euclidean time.