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

Abstract

AbstractWe focus on functional renormalization for ensembles of several (say $$n\ge 1$$ n ≥ 1 ) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form $$ \exp [-\mathrm {Tr}(V_1)\times \cdots \times \mathrm {Tr}(V_k)]$$ exp [ - Tr ( V 1 ) × ⋯ × Tr ( V k ) ] for certain noncommutative polynomials $$V_1,\ldots ,V_k\in {\mathbb {C}}_{\langle n \rangle }$$ 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 $$\mathrm U(N)$$ U ( N ) -invariants, the structure gained is the matrix algebra $$M_n( \mathcal {A}_{n,N}, \star ) $$ M n ( A n , N , ⋆ ) with entries in $$\mathcal {A}_{n,N}=({\mathbb {C}}_{\langle n \rangle } \otimes {\mathbb {C}}_{\langle n \rangle } )\oplus ( {\mathbb {C}}_{\langle n \rangle } \boxtimes {\mathbb {C}}_{\langle n \rangle })$$ A n , N = ( C ⟨ n ⟩ ⊗ C ⟨ n ⟩ ) ⊕ ( C ⟨ n ⟩ ⊠ C ⟨ n ⟩ ) , being $${\mathbb {C}}_{\langle n \rangle } $$ C ⟨ n ⟩ the free algebra generated by the n Hermitian matrices of size N (the flowing random variables) with multiplication of homogeneous elements in $$\mathcal {A}_{n,N}$$ A n , N given, for each $$P,Q,U,W\in {\mathbb {C}}_{\langle n \rangle }$$ P , Q , U , W ∈ C ⟨ n ⟩ , by $$\begin{aligned} (U \otimes W) \star ( P\otimes Q)&= PU \otimes WQ \,, \\ (U\boxtimes W) \star ( P\otimes Q)&=U \boxtimes PWQ \,, \\ (U \otimes W) \star ( P\boxtimes Q)&= WPU \boxtimes Q \,, \\ (U\boxtimes W) \star ( P\boxtimes Q)&= \mathrm {Tr} (WP) U\boxtimes Q \,, \end{aligned}$$ ( 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 $$(\lambda U) \boxtimes W = U\boxtimes (\lambda W) $$ ( λ U ) ⊠ W = U ⊠ ( λ W ) for each complex $$\lambda $$ λ , fully define the symbol $$\boxtimes $$ ⊠ .