Non-associative Algebras of Cubic Matrices and their Gauge Theories

TL;DR

The paper develops a non-associative algebra blending bimatrices and cubic matrices, organized as a two-term $L_\infty$ algebra $L^{\rm cub}_2$ with a fundamental identity linking two- and three-brackets, motivated by M-theory and the inclusion of M5-brane currents. It constructs explicit ${\rm L}^{\rm cub}_2$ algebras based on Lie algebras (e.g., $SU(2)$) and shows how gauge theories can be formulated on this structure, requiring higher-form gauge fields and yielding connections to topological BF theory and deformed IKKT models. The authors discuss several routes to consistent dynamics, including a gauge-rectifier method and background 3-forms, and outline prospects for extending to $L_3^{\rm cub}$, though challenges remain in achieving standard Yang–Mills behavior and a fermionic/supersymmetric completion. These results illuminate how non-associativity and higher-algebra structures can underpin generalized gauge theories with potential relevance to M-theory and brane dynamics. The work thus offers a concrete algebraic and field-theoretic framework for exploring non-classical gauge theories beyond conventional Lie-algebra-based YM theories, while identifying key obstacles and directions for future research.

Abstract

Motivated by M-theory, we define a new type of non-associative algebra involving usual and cubic matrices at the same time. The resulting algebra can be regarded as a two-term truncated $L_\infty$ algebra giving rise to a fundamental identity between the two- and the three-bracket. We provide a simple class of concrete examples of such algebras based on the structure constants of a Lie algebra. Connecting to previous results on higher structures, we generalize the construction of Yang-Mills theories, topological BF theory and generalized IKKT models and point out some appearing issues.