# Non-geometric Kaluza-Klein monopoles and magnetic duals of M-theory R-flux backgrounds

## TL;DR

This paper introduces a magnetic analogue of the nonassociative R-flux algebra in M-theory, relating it to monopoles, phase space structures, and non-geometric backgrounds, with implications for quantum gravity and string theory.

## Contribution

It presents a novel magnetic analogue of the R-flux algebra, relating it to Kaluza-Klein monopoles and non-geometric M-theory backgrounds, expanding understanding of phase space structures.

## Key findings

- The magnetic analogue algebra is related by a Spin(7) automorphism.
- The algebra reduces to known noncommutative and nonassociative algebras in certain limits.
- Smeared Kaluza-Klein monopoles are non-geometric and described by U(1)-gerbes.

## Abstract

We introduce a magnetic analogue of the seven-dimensional nonassociative octonionic R-flux algebra that describes the phase space of M2-branes in four-dimensional locally non-geometric M-theory backgrounds. We show that these two algebras are related by a Spin(7) automorphism of the 3-algebra that provides a covariant description of the eight-dimensional M-theory phase space. We argue that this algebra also underlies the phase space of electrons probing a smeared magnetic monopole in quantum gravity by showing that upon appropriate contractions, the algebra reduces to the noncommutative algebra of a spin foam model of three-dimensional quantum gravity, or to the nonassociative algebra of electrons in a background of uniform magnetic charge. We realise this set-up in M-theory as M-waves probing a delocalised Kaluza-Klein monopole, and show that this system also has a seven-dimensional phase space. We suggest that the smeared Kaluza-Klein monopole is non-geometric because it cannot be described by a local metric. This is the magnetic analogue of the local non-geometry of the R-flux background and arises because the smeared Kaluza-Klein monopole is described by a U(1)-gerbe rather than a U(1)-fibration.