Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes

TL;DR

This paper introduces a non-associative phase space algebra for M-theory backgrounds using octonions, explaining non-geometric fluxes and the missing momentum modes through algebraic contractions related to string coupling.

Contribution

It establishes a novel algebraic framework based on octonions to describe non-geometric M-theory backgrounds and links contraction parameters to string coupling.

Findings

Non-associative algebra of octonions models non-geometric fluxes.

Missing momentum modes are explained via algebraic contraction.

Full octonion algebra uplifted from string to M-theory context.

Abstract

We propose a non-associative phase space algebra for M-theory backgrounds with locally non-geometric fluxes based on the non-associative algebra of octonions. Our proposal is based on the observation that the non-associative algebra of the non-geometric R-flux background in string theory can be obtained by a proper contraction of the simple Malcev algebra generated by imaginary octonions. Furthermore, by studying a toy model of a four-dimensional locally non-geometric M-theory background which is dual to a twisted torus, we show that the non-geometric background is "missing" a momentum mode. The resulting seven-dimensional phase space can thus be naturally identified with the imaginary octonions. This allows us to interpret the full uncontracted algebra of imaginary octonions as the uplift of the string theory R-flux algebra to M-theory, with the contraction parameter playing the role of the string coupling constant $g_s$.