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

Abstract

Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C structure induced by an almost-complex structure \omega. Each connection A on the determinant line bundle induces a unique connection \nabla^A, and Dirac operator \D^A on spinor fields. Let \sigma: W^+ --> \Lambda^+ be the natural squaring map, taking self-dual (= positive) 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 \sigma. They are those \alpha for which (off zeros) c_1(\alpha) = c_1(\omega), where c_1(\alpha) is a suitably defined Chern class. We also obtain the formula: || \phi ||^2 D^A \phi = i (2 d^* \sigma(\phi) + < \nabla^A \phi, i \phi >)* \phi. Using these, we establish a bijective correspondence between: {Kahler forms \alpha compatible with a metric scalar-multiple of g, and with c_1(\alpha) = c_1(\omega)} and {gauge classes of pairs (\phi, A), with \nabla^A \phi = 0}, as well as a bijective correspondence between: {Symplectic forms \alpha compatible with a metric conformal to g, and with c_1(\alpha) = c_1(\omega)} and {gauge classes of pairs (\phi, A), with D^ A \phi = 0, and < \nabla^A \phi, i \phi > = 0, and \phi nowhere-zero}.