Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature

Nedim Değirmenci 1  and Şenay Karapazar 1
  • 1 Department of Mathematics, Anadolu University, Eskişehir, Turkey


Pseudo-Riemannian spinc manifolds were introduced by Ikemakhen in [7]. In the present work we consider pseudo-Riemannian 4-manifolds with neutral signature whose structure groups are SO+(2; 2). We prove that such manifolds have pseudo-Riemannian spinc structure. We construct spinor bundle S and half-spinor bundles S+ and S- on these manifolds. For the first Seiberg-Witten equation we define Dirac operator on these bundles. Due to the neutral metric self-duality of a 2-form is meaningful and it enables us to write down second Seiberg-Witten equation. Lastly we write down the explicit forms of these equations on 4-dimensional at space

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Akbulut S., Lecture Notes on Seiberg-Witten Invariants, Turkish Journal of Mathematics, 20 (1996), 95-118.

  • [2] Davidov J., Grantcharov G., Mushkarov O., Geometry of Neutral Metricin Dimension Four, arXiv:0804.2132v1.

  • [3] Değirmenci N., Özdemir N., Seiberg-Witten Equations on Lorentzianspinc manifolds, International Journal of Geometric Methods in Modern Physics, 8(4), 2011.

  • [4] Dunajki, M., West, S., Anti-Self-Dual Conformal Structures in NeutralSignature, arXiv.math/0610280v4.

  • [5] Friedrich T., Dirac Operators in Riemannian Geometry, American Mathematical Society, 2000.

  • [6] Harvey F.R., Spinors and Calibrations, Academic Press, 1990.

  • [7] Ikemakhen A., Parallel Spinors on Pseudo-Riemannian Spinc Manifolds, Journal of Geometry and Physics 9, 1473-1483, 2006.

  • [8] Kamada H., Machida Y., Self-Duality of Metrics of type (2; 2) on four dimensional manifolds, ToHoku Math. J. 49 , 259-275, 1997.

  • [9] Lawson B., Michelson M.L., Spin Geometry, Princeton University Press, 1989.

  • [10] Matsushita Y., Law P., Hitchin-Thorpe-Type Inequalities for Pseudo- Riemannian 4−Manifolds of Metric Signature (+ + −−), Geometriae Dedicata 87, 65-89, 2001.

  • [11] Matsushita Y., The existence of indefinite metrics of signature (+;+;−;−) and two kinds of almost complex structures in dimension Four, Proceedings of The Seventh International Workshop on Complex Structures and Vector Fields, Contemporary Aspects of Complex Analysis, Differential Geometry and Mathematical Physics, ed. S. Dimiev and K. Sekigawa, World Scientific, 210-225, 2005.

  • [12] Morgan J., Seiberg-Witten Equations And Applications To The Topology of Smooth Manifolds, Princeton University Press, 1996.

  • [13] Moore J., Lecture Notes on Seiberg-Witten Invariants, Springer-Verlag, 1996.

  • [14] Naber G.L., Topology, Geometry, and Gauge Fields, (Interactions), Springer-Verlag, 2011.

  • [15] Salamon D., Spin Geometry and Seiberg-Witten Invariants, 1996 (preprint).

  • [16] Witten E., Monopoles and Four Manifolds, Math. Research Letters, 1994.


Journal + Issues

The journal is published by Faculty of Mathematics and Computer Science of Ovidius University, Constanta, Romania.