The Rarita–Schwinger operator is the twisted Dirac operator restricted to \(\nicefrac 32\)-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运算符是扭曲的Dirac运算符，限于\（\ nicefrac 32 \）-spinors。Rarita-Schwinger字段是该运算符的解决方案，并且无散度。这是一个无法确定的问题，解决方案很少。对于解决方案而言，存在很大的维数空间是更加出乎意料的。在本文中，我们证明了在给定尺寸大于或等于4的情况下，存在一系列紧的流形，Rarita-Schwinger场的空间尺寸趋于无穷大。这些流形既可以简单地与具有负爱因斯坦常数的Kähler-Einstein自旋相连，又可以是具有平托托的此类空间的乘积。此外，我们用相同线性比平托环更线性独立的Rarita-Schwinger场构造甚至复杂维的Calabi-Yau流形。