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

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

2001-10-31

arxiv.org/abs/math/0110322v3

Commun. Contemp. Math., vol. 4 (2002), no. 1, pp. 45--64

Mathematics

Differential Geometry

16 pages, 2 LaTeX figures. Infinitesimal revision

Scientific paper

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}.

Affiliated with

Also associated with

No associations

Rating

Nowhere-zero harmonic spinors and their associated self-dual 2-forms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Nowhere-zero harmonic spinors and their associated self-dual 2-forms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Nowhere-zero harmonic spinors and their associated self-dual 2-forms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-105023