2016/06/27 by Varun Thakre, Thakre, Varun
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math.DG
paper · pdf · doi:10.48550/arxiv.1606.08158
15 pages, added the case of hyperKahler manifolds obtained via hyperKahler reduction and rectified a couple of embarrassing mistakes in the previous version of the paper. Comments and suggestions are welcome!
openalex publication_date 2016/06/27 · arxiv created 2016/08/24 · arxiv updated 2016/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifold. For a class of hyperKahler manifolds obtained via hyperKahler reduction, we also show that a harmonic spinor, under mild conditions, defines a symplectic structure. Finally, we show that if a covariantly constant, generalized spinor satisfies generalized Seiberg-Witten equations, the metric on the base manifold has a constant scalar curvature.