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

ABSTRACT

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 ⊠ .