nLab D=8 Seiberg-Witten theory

Idea

D = 8 Seiberg-Witten theory (short D = 8 SW) is a transfer of Seiberg-Witten theory from Riemannian 4-manifolds to Spin(7) manifolds, which are Riemannian 8-manifolds with holonomy contained in Spin(7), which leads to a canonical spinᶜ structure.

Description

For a Spin(7)-manifold, the structure group of its orientable tangent bundle can be lifted along $Spin(8)\twoheadrightarrow SO(8)$ (which is the case if and only if its second Stiefel-Whitney class vanishes) and then further reduced along the canonical inclusion $Spin(7)\hookrightarrow Spin(8)$. (It is important to note, that there are three inclusions not conjugate to each other.) With the canonical inclusion $Spin(8)\hookrightarrow Spin^\mathrm{c}(8)$, every $Spin(7)$-manifold is canonically a spinᶜ manifold. In general, not all orientable 8-manifolds are spinᶜ manifolds, making the restriction to $Spin(7)$-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

• Ayşe Hümeyra Bilge, Tekin Dereli and Şahin Koçak, Monopole equations on eight manifolds with spin(7) holonomy, Communications in Mathematical Physics 203 (1999), p. 21-30 [DOI:10.1142/S0219887812200320]

• Yi-hong Gao and Gang Tian, Instantons and the monopole-like equations in eight dimensions (2000) [arXiv:hep-th/0004167, DOI:10.1088/1126-6708/2000/05/036]

• Ayşe Hümeyra Bilge, Tekin Dereli and Şahin Koçak, Seiberg-Witten Type Monopole Equations on 8-Manifolds with Spin(7) Holonomy as Minimizers of a Quadratic Action (2003) [arXiv:hep-th/0303098]

• Şenay Karapazar Bulut, Seiberg-Witten equations on 8-dimensional manifolds with SU(4)-structure, International Journal of Geometric Methods in Modern Physics 10 (2013), [DOI:10.1142/S0219887812200320]