Manifolds with Many Rarita–Schwinger Fields

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.