Seiberg-Witten Equations on $\mathbb{R}^{6}$

Abstract

It is known that Seiberg-Witten equations are defined on smooth four dimensional manifolds. In the present work we write down a six dimensional analogue of these equations on $\mathbb{R}^{6}$. To express the first equation, the Dirac equation, we use a unitary representation of complex Clifford algebra $\mathbb{Cl}_{2n}$. For the second equation, a kind of self-duality concept of a two-form is needed, we make use of the decomposition $\Lambda^{2}(\mathbb{R}^{6}) = \Lambda^{2}_{1}(\mathbb{R}^6) \oplus \Lambda^{2}_{6}(\mathbb{R}^6) \oplus \Lambda^{2}_{8}(\mathbb{R}^6)$. We consider the eight-dimensional part $\Lambda^{2}_{8}(\mathbb{R}^6)$ as the space of self-dual two-forms.