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. 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.

Anahtar Kelimeler:

Seiberg--Witten equations, spinor, Dirac operator, contact metric manifold, self-duality

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

Keywords:

Seiberg--Witten equations, spinor, Dirac operator, contact metric manifold, self-duality,

PDF

___

eOne can show that∇Ai0 ψ0= 0 . Therefore, we deduce that DAψ = 0. H Moreover, DAψ = 0. The pair (A, ψ = −sHψ0) is a solution of Seiberg–Witten-like equations in (3). Akbulut S. Lectures on Seiberg-Witten Invariants. Turk J Math 1996; 20: 95–118.
Baum H. Lorentzian twistor spinors and CR-geometry. Differ Geom Appl 1996; 11: 69–96.
Bellettini C. Almost complex structures and calibrated integral cycles in contact 5-manifolds. Adv Calc Var 2013; 6: 339–374.
Bilge AH, Dereli T, Ko¸cak S¸. Monopole equations on 8-manifolds with Spin(7) holonomy. Commun Math Phys 1999; 203: 21–30.
Corrigan E, Devchand C, Fairlie DB, Nuyts J. First-order equations for gauge fields in spaces of dimension greater than four. Nucl Phys B 1983; 214: 452–464.
Degirmenci N, Karapazar S¸. Seiberg-Witten like monopole equations onR5. J Partial Dif 2011; 24: 150–157.
Degirmenci N, ¨Ozdemir N. Seiberg-Witten like equations on 7-manifolds with G2-structures. J Nonlinear Math Phys 2005; 12: 457–461.
Fan H. Half De Rham complexes and line fields on odd-dimensional manifolds. Trans Am Math Society 1996; 348: 2947–2982.
Friedrich T. Dirac Operators in Riemannian Geometry. Providence, RI, USA: AMS, 2000.
Gao YH, Tian G. Instantons and the monopole-like equations in eight dimensions. J High Energy Phys 2000; 5: 0
Morgan J. Seiberg-Witten Equations and Applications to the Topology of Smooth Manifolds. Princeton, NJ, USA: Princeton University Press, 1996.
Petit R. Spinc-structures and Dirac operators on contact manifolds. Differ Geom Appl 2005; 22: 229–252.
Tian G. Gauge theory and calibrated geometry I. Ann Math 2000; 151: 193–268.
Witten E. Monopoles and four manifolds. Math Res Lett 1994; 1: 769–796.