We study the one loop renormalisation of 4d SU(N) Yang-Mills theory with M adjoint representation scalar multiplets related by O(M) symmetry. General M are of field theoretic interest, and the 4d one loop beta function of the gauge coupling g² vanishes for the case M = 22, which is intriguing for string theory. This case is related to D3 branes of critical bosonic string theory in D = 22 + 4 = 26. An RG fixed point could have provided a definition for a purely bosonic AdS/CFT, but we show that scalar self-couplings λ ruin one-loop conformal invariance in the large N limit. There are real fixed flows (fixed points of λ/g²) only for M ≥ 406, rendering one-loop fixed points of the gauge coupling and scalar couplings incompatible.

We develop and check an algebraic approach to the one-loop renormalisation group which we find to be characterised by a non-associative algebra of marginal couplings. In the large N limit, the resulting RG flows typically suffer from strong coupling in both the ultraviolet and the infrared. Only for M ≥ 406 fine-tuned solutions exist which are weakly coupled in the infrared.

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.

We apply our previously developed approach to marginal quartic interactions in multiscalar QFTs, which shows that one-loop RG flows can be described in terms of a commutative algebra, to various models in 4d. We show how the algebra can be used to identify optimal scalings of the couplings for taking large N limits. The algebra identi- fies these limits without diagrammatic or combinatorial analysis. For several models this approach leads to new limits yet to be explored at higher loop orders. We consider the bifundamental and trifundamental models, as well as a matrix-vector model with an ad- joint representation. Among the suggested new limit theories are some which appear to be less complex than general planar limits but more complex than ordinary vector models or melonic models.