Manifolds with Many Rarita–Schwinger Fields

Abstract

The Rarita–Schwinger operator is the twisted Dirac operator restricted to $$\nicefrac 32$$ 3 2 -spinors. Rarita–Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita–Schwinger fields tends to infinity. These manifolds are either simply connected Kähler–Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi–Yau manifolds of even complex dimension with more linearly independent Rarita–Schwinger fields than flat tori of the same dimension.

Rarita-Schwinger fields on nearly Kähler manifolds

“…The ∇-parallel complex volume form is represented as ψ + + iψ − . Also, we know the (real) volume form vol = vol g coincides with 1 4 ψ + ∧ ψ − . It is a prominent property that the pair (ω, ψ + ) leads to an SU(3)-structure on T M (cf.…”

“…Since the scalar curvature is normalized as scal = 30, M admits a unit Killing spinor κ with the Killing number 1 2 . The Killing spinor κ defines a bundle map γ → γ • κ from ∧ 0 M ⊕∧ 1 M ⊕∧ 6 M to S 1/2 .…”

“…So, the Killing spinor κ 0 is corresponding to the nearly Kähler structure (g, J). For (i) in the above definition, we take κ as a constant multiplication of κ 0 because the space of the Killing spinors with the Killing number 1 2 is one-dimensional. We adjust the symbols to the ones we used in Section 4.…”

“…The spectrum of the Rarita-Schwinger operator on some concrete symmetric spaces was computed by Homma and Tomihisa [12]. Bär and Mazzeo [1] proved that there exists a compact manifold with many Rarita-Schwinger fields in any given dimension n ≥ 4. In particular, if n is divisible by 4, then we can take this manifold as a simply connected compact manifold with negative Einstein constant.…”

“…We study deformations of Killing spinors on nearly Kähler manifolds. Let E(λ) be the λ-eigenspace of the Laplace operator restricted to co-closed primitive (1, 1)-forms, and K + be the space of the Killing spinors with the Killing number 1 2 . Then our result in Section 5 is Theorem B.…”