Instanton

Remarks

Some authors use the opposite orientation and call self-dual connections, with $\star F_A = F_A$, instantons. Anti-self-dual connections satisfy the Yang–Mills equations $d_A \star F_A = 0$, because $d_A \star F_A = -d_A F_A = 0$ by the Bianchi identity. On a compact $X$, they minimize the Yang–Mills functional $\int_X \abs{F_A}^2$ among connections on $P$, the minimum being a topological constant times the absolute value of a characteristic number of $P$.

Example: a flat connection, with $F_A = 0$, is an instanton. The BPST instanton on $\R^4$ for $G = SU(2)$ has curvature that is nowhere zero and anti-self-dual for one choice of orientation.

Non-example: on the trivial $U(1)$-bundle over $\R^4$ with coordinates $x_1, \dots, x_4$, let $A = i \, x_1 \, dx_2$. Then $F_A = i \, dx_1 \wedge dx_2$ and $\star F_A = i \, dx_3 \wedge dx_4 \ne -F_A$, so $A$ is not an instanton.