Harmonic Spinors

σk-Scalar curvature and eigenvalues of the Dirac operator

“…A proof of this can be found in [8] or in [14]. It is easy to show that µ 2 = F 2 (g)/vol(g) by integrating (16) …”

“…On a product manifold M 4 = Σ 2 1 × Σ 2 2 of two surfaces Σ 2 1 and Σ 2 2 of constant negative Gaussian curvature, one can find a spin structure such that the corresponding Dirac operator admits harmonic spinors (cf. [16]). It is clear that its σ 2 is a positive constant.…”

(0,4) dualities

“…The number of chiral multiplets of the N = (2, 2) twist (or SU(2) I twist) is given by the number of harmonic spinors on the curve C g or h 0 (C g , K 1 2 ). This number depends on the choice of spin structure on C g [44].…”

Smooth structures on complex surfaces with fundamental group $\mathbf {Z}_2$

“…Suppose otherwise that X has a spin structure, which can then be pulled back to a σ-equivariant spin structure on X. As spin structures on X are in one-one canonical correspondence with the (holomorphic) square roots of the line bundle K X by Hitchin [14], this means that σ can be lifted to an involution on one of the square roots and consequently can be lifted to a bundle isomorphism on K X . In particular σ * c 1 (K X ) = c 1 (K X ) for the Chern class of K X .…”