Higher Quantum Geometry and Non-Geometric String Theory

We present a concise overview of the physical and mathematical structures underpinning the appearence of nonassociative deformations of geometry in non-geometric string theory. Starting from a quick recap of the appearence of noncommutative product and commutator deformations of geometry in open string theory with $B$-fields, we argue on physical principles that closed strings should instead probe triproduct and tribracket deformations in backgrounds of locally non-geometric fluxes. After describing the toy model of electric charges moving in fields of smooth distributions of magnetic charge as a physical introduction to the notions of nonassociative geometry, we review the description of non-geometric fluxes in generalized geometry and double field theory, and the worldsheet calculations suggesting the appearence of nonassociative deformations, together with their caveats. We discuss how algebroids and their associated AKSZ sigma-models give a description of non-geometric backgrounds in terms of higher geometry, and consider the quantization of the membrane sigma-model which geometrizes closed strings with $R$-flux. From this we derive an explicit nonassociative star product for the quantum geometry of the closed string phase space, and apply it to derive the triproducts that appear in conformal field theory correlation functions, to describe a consistent treatment of nonassociative quantum mechanics, to demonstrate quantitatively the coarse-graining of spacetime due to $R$-flux, and to describe the quantization of Nambu brackets. We also briefly review how these constructions lead to a nonassociative theory of gravity, their uplifts to non-geometric M-theory, and the role played by $L_\infty$-algebras in these developments.