Self-dual differential form (${\Omega }_{+}^2$)

On an oriented Riemannian four-manifold, a two-form $\alpha$ is self-dual when $\ast \alpha =\alpha$.

On an oriented Riemannian four-manifold, a two-form is self-dual when $\ast \alpha =\alpha$. Here $\ast$ is the Hodge star operator.

For the orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$, define ${\eta }_{ij}^a={ϵ}_{aij}$, ${\eta }_{i4}^a={\delta }_{ai}$, and extend antisymmetrically. The three forms $\frac{1}{2}{\eta }_{\mu \nu }^{a}d{x}^{\mu }\wedge d{x}^{\nu }$ are self-dual. Reversing the sign of the components involving index four gives the anti-self-dual symbols $\bar{\eta }$. They identify the two three-dimensional eigenspaces of the Hodge star on differential two-forms and give compact formulas for the BPST instanton.

For $w=x^1+i{x}^2$, $z=x^3+i{x}^4$ and the standard oriented Euclidean metric, the real and imaginary parts of $dw\wedge dz$, together with $i(dw\wedge d\bar{w}+dz\wedge d\bar{z})$, span the self-dual two-forms. The printed complex four-form $dw\wedge dz\wedge d\bar{w}\wedge d\bar{z}$ is four times the metric volume form and gives the same orientation.