Nonassociativity, Malcev Algebras and String Theory

TL;DR

This paper explores the role of nonassociative Malcev algebras in string theory, showing their connection to nonassociative structures in D-branes, quantum mechanics, and string field theory, suggesting a new perspective on non-linear quantum mechanics with a fundamental length.

Contribution

It identifies nonassociative Malcev algebras as a unifying structure in string theory, quantum mechanics, and string field theory, extending their applications to include electric and magnetic charges.

Findings

Nonassociative structures in string backgrounds relate to Malcev algebras.

Generalization of Malcev algebras includes electric and magnetic charges.

Connection between nonassociative string theory and non-linear quantum mechanics.

Abstract

Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality.