• DOI:10.1007/s11005-022-01546-x
• Corpus ID: 242757365

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

@article{PrezSnchez2021ARG,
  title={A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions},
  author={Carlos I. P{\'e}rez-S{\'a}nchez},
  journal={Letters in Mathematical Physics},
  year={2021},
  volume={112}
}

• Carlos I. Pérez-Sánchez
• Published 4 November 2021
• Materials Science
• Letters in Mathematical Physics

We focus on functional renormalization for ensembles of several (say n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form exp[-Tr(V1)×⋯×Tr(Vk)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \exp [-\mathrm {Tr}(V_1)\times \cdots \times \mathrm {Tr}(V_k)]$$\end{document} for certain noncommutative polynomials V1,…,Vk∈C⟨n⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1,\ldots ,V_k\in {\mathbb {C}}_{\langle n \rangle }$$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm U(N)$$\end{document}-invariants, the structure gained is the matrix algebra Mn(An,N,⋆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_n( \mathcal {A}_{n,N}, \star ) $$\end{document} with entries in An,N=(C⟨n⟩⊗C⟨n⟩)⊕(C⟨n⟩⊠C⟨n⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_{n,N}=({\mathbb {C}}_{\langle n \rangle } \otimes {\mathbb {C}}_{\langle n \rangle } )\oplus ( {\mathbb {C}}_{\langle n \rangle } \boxtimes {\mathbb {C}}_{\langle n \rangle })$$\end{document}, being C⟨n⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}_{\langle n \rangle } $$\end{document} the free algebra generated by the n Hermitian matrices of size N (the flowing random variables) with multiplication of homogeneous elements in An,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_{n,N}$$\end{document} given, for each P,Q,U,W∈C⟨n⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P,Q,U,W\in {\mathbb {C}}_{\langle n \rangle }$$\end{document}, 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 (λU)⊠W=U⊠(λW)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\lambda U) \boxtimes W = U\boxtimes (\lambda W) $$\end{document} for each complex λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, fully define the symbol ⊠\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boxtimes $$\end{document}.