nLab D=7 Seiberg-Witten theory

Idea

D = 7 Seiberg-Witten theory (short D = 7 SW) is a transfer of Seiberg-Witten theory from Riemannian 4-manifolds to G₂ manifolds, which are Riemannian 7-manifolds with holonomy contained in G₂, which leads to a canonical spinᶜ structure.

Description

For a G₂ manifold, the structure group of its orientable tangent bundle can be lifted along $Spin(7)\twoheadrightarrow SO(7)$ (which is the case if and only if its second Stiefel-Whitney class vanishes) and then further reduced along the canonical inclusion $G_2\hookrightarrow Spin(7)$. With the canonical inclusion $Spin(7)\hookrightarrow Spin^\mathrm{c}(7)$, every $G_2$-manifold is canonically a spinᶜ manifold. In general, not all orientable 7-manifolds are spinᶜ manifolds, making the restriction to $G_2$-manifolds necessary. Since all orientable 4-manifolds are spinᶜ manifolds, a similar restriction is not necessary in usual Seiberg-Witten theory.

Related concepts

Articles about Seiberg-Witten theory:

• Ordinary theory: Seiberg-Witten equations, linearized Seiberg-Witten equations, Seiberg-Witten invariant, Seiberg-Witten moduli space, Seiberg-Witten flow
• Adapted theory: D=6 Seiberg-Witten theory, D=7 Seiberg-Witten theory (over G₂ manifolds), D=8 Seiberg-Witten theory (over Spin(7) manifolds), quaternionic Seiberg-Witten theory, pseudo-Riemannian Seiberg-Witten theory

References

• Nedim Değirmenci and Nülifer Özdemir, Seiberg-Witten-like Equations on 7-Manifolds with G₂-Structure, Journal of Nonlinear Mathematical Physics 12 4 (2005), p. 457–461