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 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.