Hidden symmetry in the presence of fluxes

TL;DR

This paper derives the most general first order symmetry operator for the Dirac equation with arbitrary fluxes, revealing new constraints and special cases relevant to physical theories involving fluxes and torsion.

Contribution

It introduces a generalized conformal Killing-Yano system that characterizes symmetry operators for the Dirac equation with fluxes, extending previous results and analyzing special physical cases.

Findings

Decouples into known conformal Killing-Yano equations with torsion

Identifies symmetry operators in flux and torsion backgrounds

Analyzes Dirac equations with scalar, 5-form, and 7-form fluxes

Abstract

We derive the most general first order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogenous form omega which is a solution to a coupled system of first order partial differential equations which we call the generalized conformal Killing-Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U(1) field, the system of generalized conformal Killing-Yano equations decouples into the homogenous conformal Killing-Yano equations with torsion introduced in [arXiv:0905.0722] and the symmetry operator is essentially the one derived in [arXiv:1002.3616]. We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.