TCFHs, hidden symmetries and M-theory backgrounds

TL;DR

This paper investigates the symmetries and integrability properties of supersymmetric solutions in 11-dimensional supergravity, revealing hidden symmetries, conserved charges, and the role of form bilinears in particle probe dynamics.

Contribution

It generalizes the conformal Killing-Yano equation for supergravity solutions and analyzes the integrability and hidden symmetries of various M-theory backgrounds.

Findings

Form bilinears satisfy a generalized conformal Killing-Yano equation.

All spherically symmetric M-branes have completely integrable geodesic flows.

Form bilinears generate hidden symmetries for spinning particle probes.

Abstract

We present the TCFH of 11-dimensional supergravity and so demonstrate that the form bilinears of supersymmetric solutions satisfy a generalisation of the conformal Killing-Yano equation with resepct to the TCFH connection. We also compute the Killing-St\"ackel, Killing-Yano and closed conformal Killing-Yano tensors of all spherically symmetric M-branes that include the M2-brane, M5-brane, KK-monopole and pp-wave and demonstrate that their geodesic flows are completely integrable by giving all independent conserved charges in involution. We then find that all form bilinears of pp-wave and KK-monopole solutions generate (hidden) symmetries for spinning particle probes propagating on these backgrounds. Moreover, there are Killing spinors such that some of the 1-, 2- and 3-form bilinears of the M2-brane solution also generate symmetries for spinning particle probes. We also explore the question on whether the form bilinears are sufficient to prove the integrability of particle probe dynamics on 11-dimensional supersymmetric backgrounds.