Nowhere-zero harmonic spinors and their associated self-dual 2-forms

Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin(C)-structure induced by an almost-complex structure w. Each connection A on the determinant line bundle induces a unique connection del(A), and Dirac operator D-A on spinor fields. Let sigma : W+ --> A(+) be the natural squaring map, taking self-dual spinors to self-dual 2-forms. In this paper, we characterize the self-dual 2-forms that are images of self-dual spinor fields through alpha. They are those a for which (off zeros) c(1)(alpha) = c(1)(w), where c(1)(alpha) is a suitably defined Chern class. We also obtain the formula: \\rho\\(2)D(A)rho = i(2d*sigma(rho) + <del(A)rho,irho>(R))(.)rho. Using these, we establish a bijective correspondence between: {Kahler forms alpha compatible with a metric scalax-multiple of g, and with c(1)(alpha) = c(1)(w)} and {gauge classes of pairs (rho,A), with del(A)rho = 0}, as well as a bijective correspondence between. {symplectic forms alpha compatible with a metric conformal to g, and with c(1)(a) = c(1)(w)} and {gauge classes of pairs (rho, A), with D(A)rho = 0, and <del(A)rho,irho>(R) = 0, and rho nowhere-zero}.