Non-associative star products and quantization of non-geometric backgrounds in string and M-theory

We review two known in the literature exemples of non-associative star products. The first one is the phase space star product representing quantization of non-geometric $R$-flux background in closed string theory. The second is the octonionic star product which provides the quantization of the quasi-Poisson algebra isomorphic to the Malcev algebra of imaginary octonions. We discuss in details the construction, properties and physical applications of these star products. In particular, we consider the quantization of non-geometric M-theory background. Based on these two exemples we formulate the minimal set of physically motivated conditions for the definition of non-associative deformation quantization.