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 nge 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)] for certain noncommutative polynomials V_1,ldots ,V_kin {mathbb {C}}_{langle n rangle } 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)-invariants, the structure gained is the matrix algebra M_n( mathcal {A}_{n,N}, star ) 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 }), being {mathbb {C}}_{langle n rangle } 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} given, for each P,Q,U,Win {mathbb {C}}_{langle n rangle }, by (U⊗W)⋆(P⊗Q)=PU⊗WQ,(U⊠W)⋆(P⊗Q)=U⊠PWQ,(U⊗W)⋆(P⊠Q)=WPU⊠Q,(U⊠W)⋆(P⊠Q)=Tr(WP)U⊠Q,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (U \\otimes W) \\star ( P\\otimes Q)&= PU \\otimes WQ \\,, \\\\ (U\\boxtimes W) \\star ( P\\otimes Q)&=U \\boxtimes PWQ \\,, \\\\ (U \\otimes W) \\star ( P\\boxtimes Q)&= WPU \\boxtimes Q \\,, \\\\ (U\\boxtimes W) \\star ( P\\boxtimes Q)&= \\mathrm {Tr} (WP) U\\boxtimes Q \\,, \\end{aligned}$$\\end{document}which, together with the condition (lambda U) boxtimes W = Uboxtimes (lambda W) for each complex lambda , fully define the symbol boxtimes .