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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.

We quantise a Poisson structure on H^{n+2g}, where H is a semidirect product group of the form $G\ltimes\mathfrak{g}^*$. This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$ on $R \times S_{g,n}$, where S_{g,n} is a surface of genus g with n punctures. The quantisation of this Poisson structure is a key step in the quantisation of Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$. We construct the quantum algebra and its irreducible representations and show that the quantum double D(G) of the group G arises naturally as a symmetry of the quantum algebra.

We consider spaces of "virtual" constrained generalized Killing spinors, i.e. spaces of Majorana spinors which correspond to "off-shell" $s$-extended supersymmetry in compactifications of eleven-dimensional supergravity based on eight-manifolds $M$. Such spaces naturally induce two stratifications of $M$, called the chirality and stabilizer stratification. For the case $s=2$, we describe the former using the canonical Whitney stratification of a three-dimensional semi-algebraic set ${\cal R}$. We also show that the stabilizer stratification coincides with the rank stratification of a cosmooth generalized distribution ${\cal D}_0$ and describe it explicitly using the Whitney stratification of a four-dimensional semi-algebraic set $\mathfrak{P}$. The stabilizer groups along the strata are isomorphic with $\mathrm{SU}(2)$, $\mathrm{SU}(3)$, $\mathrm{G}_2$ or $\mathrm{SU}(4)$, where $\mathrm{SU(2)}$ corresponds to the open stratum, which is generically non-empty. We also determine the rank stratification of a larger generalized distribution ${\cal D}$ which turns out to be integrable in the case of compactifications down to $\mathrm{AdS}_3$.