New quantum numbers for the Dirac equation in curved spacetime

We show that, on spacetimes which admit Yano tensors, it is possible to construct operators that (anti)commute with the Dirac operator, thus providing extra quantum numbers even when isometries are not present. This is the main result obtained and is valid for Yano tensors of arbitrary rank. It implies that the theory of the spinning particle in such spacetimes has no anomalies and admits genuine quantum mechanical extra supersymmetries. If a Killing spinor is present, that is, the spacetime has almost special holonomy, then it is possible to construct a tower of Yano tensors of different rank from it. We give a full description of this structure and its relation to Hodge duality and the conformal Yano equation. As a concrete application of our result, we construct Yano operators on maximally symmetric spacetimes, where the underlying group structure greatly simplifies the calculation. The high symmetry tightly constrains the form of Yano tensors in these spacetimes: they are spanned by tensor products of Killing vectors.