nLab D=6 Seiberg-Witten theory

Idea

D = 6 Seiberg-Witten theory (short D = 6 SW) is a transfer of Seiberg-Witten theory from Riemannian 4-manifolds to Riemannian 6-manifolds.

Description

While usual Seiberg-Witten theory is based on the Lie group $Spin^\mathrm{c}(4)=U(2)\times_{U(1)}U(2)$, D=6 Seiberg-Witten theory is based on the Lie group:

$$Spin^\mathrm{c}(6) \coloneqq\big(Spin(6)\times U(1)\big)/\mathbb{Z}_2 \cong\big(SU(4)\times U(1)\big)/\mathbb{Z}_2.$$

With $U(4)\cong\big(SU(4)\times U(1)\big)/\mathbb{Z}_4$ one has a double cover $Spin^\mathrm{c}(6)\twoheadrightarrow U(4)$.

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 Şenay Karapazar Bulut, Seiberg-Witten-like equations on 6-dimensional SU(3)-manifolds (2010) [pdf]

• Yuuji Tanaka, Seiberg-Witten type equations on compact symplectic 6-manifolds (2014) [arXiv:1407.1934]