A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions.

We focus on functional renormalization for ensembles of several (say n ≥ 1 ) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form exp [ - Tr ( V 1 ) × â‹¯ × Tr ( V k ) ] for certain noncommutative polynomials V 1 , … , V k ∈ C ⟨ n ⟩ in the n matrices. This article shows how the "algebra of functional renormalization"-that is, the structure that makes the renormalization flow equation computable-is derived from ribbon graphs, only by requiring the one-loop structure that such equation (due to Wetterich) is expected to have. Whenever it is possible to compute the renormalization flow in terms of U ( N ) -invariants, the structure gained is the matrix algebra M n ( A n , N , ⋆ ) with entries in A n , N = ( C ⟨ n ⟩ ⊗ C ⟨ n ⟩ ) ⊕ ( C ⟨ n ⟩ ⊠ C ⟨ n ⟩ ) , being C ⟨ n ⟩ the free algebra generated by the n Hermitian matrices of size N (the flowing random variables) with multiplication of homogeneous elements in A n , N given, for each P , Q , U , W ∈ C ⟨ n ⟩ , by ( U ⊗ W ) ⋆ ( P ⊗ Q ) = P U ⊗ W Q , ( U ⊠ W ) ⋆ ( P ⊗ Q ) = U ⊠ P W Q , ( U ⊗ W ) ⋆ ( P ⊠ Q ) = W P U ⊠ Q , ( U ⊠ W ) ⋆ ( P ⊠ Q ) = Tr ( W P ) U ⊠ Q , which, together with the condition ( λ U ) ⊠ W = U ⊠ ( λ W ) for each complex λ , fully define the symbol ⊠ .

On Multimatrix Models Motivated by Random Noncommutative Geometry II: A Yang-Mills-Higgs Matrix Model.

We continue the study of fuzzy geometries inside Connes' spectral formalism and their relation to multimatrix models. In this companion paper to Pérez-Sánchez (Ann Henri Poincaré 22:3095-3148, 2021, arXiv:2007.10914), we propose a gauge theory setting based on noncommutative geometry, which-just as the traditional formulation in terms of almost-commutative manifolds-has the ability to also accommodate a Higgs field. However, in contrast to 'almost-commutative manifolds', the present framework, which we call gauge matrix spectral triples, employs only finite-dimensional algebras. In a path-integral quantization approach to the Spectral Action, this allows to state Yang-Mills-Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields exactly mirror those of the Yang-Mills-Higgs theory on a smooth manifold.