Nowhere-zero harmonic spinors and their associated self-dual 2-forms
Abstract
Description
Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection \nabla^A, and Dirac operator \D^A on spinor fields. Let σ: W^+ --> Λ^+ 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 σ. They are those αfor which (off zeros) c_1(α) = c_1(ω), where c_1(α) is a suitably defined Chern class. We also obtain the formula: || ϕ||^2 D^A ϕ= i (2 d^* σ(ϕ) + < \nabla^A ϕ, i ϕ>)* ϕ.
Using these, we establish a bijective correspondence between: {Kahler forms αcompatible with a metric scalar-multiple of g, and with c_1(α) = c_1(ω)} and {gauge classes of pairs (ϕ, A), with \nabla^A ϕ= 0}, as well as a bijective correspondence between: {Symplectic forms αcompatible with a metric conformal to g, and with c_1(α) = c_1(ω)} and {gauge classes of pairs (ϕ, A), with D^ A ϕ= 0, and < \nabla^A ϕ, i ϕ> = 0, and ϕnowhere-zero}.
16 pages, 2 LaTeX figures. Infinitesimal revision
16 pages, 2 LaTeX figures. Infinitesimal revision