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Seiberg–Witten-like equations on 5-dimensional contact metric manifolds

In this paper, we write Seiberg–Witten-like equations on contact metric manifolds of dimension 5. Since any contact metric manifold has a Spinc-structure, we use the generalized Tanaka–Webster connection on a Spinc spinor bundle of a contact metric manifold to define the Dirac-type operators and write the Dirac equation. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 812 – 818 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1303-34 Seiberg–Witten-like equations on 5-dimensional contact metric manifolds ∗∗ ˇ IRMENC ˙ ˙ S Nedim DEG I, ¸ enay BULUT∗ Department of Mathematics, Faculty of Science, Anadolu University, Eski¸sehir, Turkey Received: 18.03.2013 • Accepted: 26.02.2014 • Published Online: 01.07.2014 • Printed: 31.07.2014 Abstract: In this paper, we write Seiberg–Witten-like equations on contact metric manifolds of dimension 5 . Since any contact metric manifold has a Spin c -structure, we use the generalized Tanaka–Webster connection on a Spin c spinor bundle of a contact metric manifold to define the Dirac-type operators and write the Dirac equation. The self-duality of 2 -forms needed for the curvature equation is defined by using the contact structure. These equations admit a nontrivial solution on 5 -dimensional strictly pseudoconvex CR manifolds whose contact distribution has a negative constant scalar curvature. Key words: Seiberg–Witten equations, spinor, Dirac operator, contact metric manifold, self-duality 1. Introduction Seiberg–Witten equations were defined on 4-dimensional Riemannian manifolds by Witten in [14]. The solution space of these equations gives differential topological invariants for 4-manifolds [1, 11]. Some generalizations were given later on higher dimensional manifolds [4, 7, 10]. Seiberg–Witten equations consist of 2 equations. The first is the Dirac equation, which is meaningful for the manifolds having Spin c −structure. The second is the curvature equation, which couples the self-dual part of a connection 2-form with a spinor field. In order to be able to write down the curvature equation, the notion of the self-duality of a 2-form is needed. This notion is meaningful for 4 -dimensional Riemannian manifolds. On the other hand, there are similar self-duality notions for some higher .