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

TL;DR

This paper reviews known non-associative star products in string and M-theory, focusing on their construction, properties, and physical applications, especially in non-geometric backgrounds.

Contribution

It introduces a minimal set of physically motivated conditions for non-associative deformation quantization based on two key examples.

Findings

Quantization of non-geometric R-flux background in string theory.

Octonionic star product for Malcev algebra of octonions.

Formulation of conditions for non-associative deformation quantization.

Abstract

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.