Seiberg-Witten equations are usually studied on 4- and 3-dimensional manifolds. The equation can also be formulated on 8-dimensional manifolds with holonomy contained in \(\mathrm{Spin}(7)\), see [A. Bilge, Commun. Math. Phys. 203, No. 1, 21–30 (1999; Zbl 0946.58028)] and [Y. H. Gao and G. Tian, J. High Energy Phys. 4, No. 5B, Paper No.05(2000)036, 22 p. (2000; Zbl 0990.81550)]. In the article under review the special case of holonomy contained in \(\mathrm{SU}(4)\subset \mathrm{Spin}(7)\) is studied. Holonomy in \(\mathrm{SU}(4)\) is equivalent to saying that we have a Calabi-Yau manifold, i.e. a Ricci-flat Kähler manifold, in dimension \(8\). In this case the Seiberg-Witten equations can be expressed with the help of the complex structure, similar to the Seiberg-Witten equations in \(4\) dimensions where the complex structure of a Kähler surface simplifies the equations.