Quantum Mechanics of Yano tensors: Dirac equation in curved spacetime

TL;DR

This paper demonstrates that Yano tensors of any rank in curved spacetime generate non-anomalous quantum symmetries of the Dirac equation, linking classical conserved quantities to quantum operators and exploring their relation to supergravity.

Contribution

It extends the known classical and quantum correspondence of Yano tensor symmetries from rank-two to arbitrary rank, and explores their invariance and relation to supergravity and maximally symmetric spaces.

Findings

Yano symmetries are preserved at the quantum level for all ranks.

Yano tensors are invariant under Hodge duality.

Construction of Yano tensors on maximally symmetric spaces using Killing vectors.

Abstract

In spacetimes admitting Yano tensors the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to operators that in the classical limit commute with the Dirac operator and generate conserved quantities. We show that the result is preserved in the full quantum theory, that is, Yano symmetries are not anomalous. This was known for Yano tensors of rank two, but our main result is to show that it extends to Yano tensors of arbitrary rank. We also describe the conformal Yano equation and show that is invariant under Hodge duality. There is a natural relationship between Yano tensors and supergravity theories. As the simplest possible example, we show that when the spacetime admits a Killing spinor then this generates Yano and conformal Yano tensors. As an application, we construct Yano tensors on maximally symmetric spaces: they are spanned by tensor products of Killing vectors.