# $G_2$-structures and quantization of non-geometric M-theory backgrounds

## TL;DR

This paper develops a quantization framework for non-geometric M-theory backgrounds using $G_2$-structures, leading to nonassociative star products and triproducts that connect M-theory and string theory models.

## Contribution

It introduces a novel deformation quantization approach based on $G_2$-structures, extending to $Spin(7)$-structures, and links nonassociative algebraic structures to M-theory backgrounds.

## Key findings

- Derived a phase space star product for nonassociative M-theory backgrounds.
- Showed reduction of the star product to string theory $R$-flux models.
- Proposed a 3-algebra structure on M2-brane phase space.

## Abstract

We describe the quantization of a four-dimensional locally non-geometric M-theory background dual to a twisted three-torus by deriving a phase space star product for deformation quantization of quasi-Poisson brackets related to the nonassociative algebra of octonions. The construction is based on a choice of $G_2$-structure which defines a nonassociative deformation of the addition law on the seven-dimensional vector space of Fourier momenta. We demonstrate explicitly that this star product reduces to that of the three-dimensional parabolic constant $R$-flux model in the contraction of M-theory to string theory, and use it to derive quantum phase space uncertainty relations as well as triproducts for the nonassociative geometry of the four-dimensional configuration space. By extending the $G_2$-structure to a $Spin(7)$-structure, we propose a 3-algebra structure on the full eight-dimensional M2-brane phase space which reduces to the quasi-Poisson algebra after imposing a particular gauge constraint, and whose deformation quantisation simultaneously encompasses both the phase space star products and the configuration space triproducts. We demonstrate how these structures naturally fit in with previous occurences of 3-algebras in M-theory.