nLab pseudo-Riemannian Seiberg-Witten theory

Idea

Pseudo-Riemannian Seiberg-Witten theory is the transfer of Seiberg-Witten theory from Riemannian 4-manifolds to pseudo-Riemannian 4-manifolds.

Description

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

$$Spin_+^\mathrm{c}(2,2) \coloneqq\big(Spin_+(2,2)\times U(1)\big)/\mathbb{Z}_2 \cong U(1,1)\times_{\operatorname{U}(1)}\operatorname{U}(1,1) =\{(A,B)\in\operatorname{U}(1,1)\times\operatorname{U}(1,1)|\det(A)=\det(B)\}.$$

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 equations on four-dimensional Lorentzian spinc manifolds (2011), [DOI:10.1142/S0219887811005348]

• Nedim Değirmenci and Şenay Karapazar, Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature, Analele ştiinţifice ale Universităţii “Ovidius” Constanţa. Seria Matematică 20 1 (2012)