Classifying large N limits of multiscalar theories by algebra

We develop a new approach to RG flows and show that one-loop flows in multiscalar theories can be described by commutative but non-associative algebras. As an example related to $D$-brane field theories and tensor models, we study the algebra of a theory with $M$ $SU(N)$ adjoint scalars and its large $N$ limits. The algebraic concepts of idempotents and Peirce numbers/Kowalevski exponents are used to characterise the RG flows. We classify and describe all large $N$ limits of algebras of multiadjoint scalar models: the standard `t Hooft matrix theory limit, a `multi-matrix' limit, each with one free parameter, and an intermediate case with extra symmetry and no free parameter of the algebra, but an emergent free parameter from a line of one-loop fixed points. The algebra identifies these limits without diagrammatic or combinatorial analysis.