Understand
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.